MIKE ———————————————————————-

Slide 1. Slide 2. Slide 3. Slide 4. Slide 5. Slide 6. Slide 7. Slide 8. Slide 9. Slide 10. Slide 11.

JOCELYN ——————————————————————-

Clustering: K-Means

Definition and Overview

K-means is a simple unsupervised machine learning algorithm for clustering. From the textbook, the algorithm is as follows:

  • Select K centroids (K rows chosen at random).
  • Assign each data point to its closest centroid.
  • Recalculate the centroids as the average of all data points in a cluster (that is, the centroids are p-length mean vectors, where p is the number of variables).
  • Assign data points to their closest centroids.
  • Continue steps 3 and 4 until the observations aren’t reassigned or the maximum number of iterations (R uses 10 as a default) is reached.

The distances are calculated by a squared error function.

\[\sum_{j=1}^{k} \sum_{i=1}^{n}||x_i^{(j)} - c_j||^2\]

Note that we need to know the number of clusters beforehand to be able to calculate the centroids.

Example

Let’s look at an example with K-Means. For easier visualization, we’ll use a pretty simple dataset built into R, iris.

str(iris)
## 'data.frame':    150 obs. of  5 variables:
##  $ Sepal.Length: num  5.1 4.9 4.7 4.6 5 5.4 4.6 5 4.4 4.9 ...
##  $ Sepal.Width : num  3.5 3 3.2 3.1 3.6 3.9 3.4 3.4 2.9 3.1 ...
##  $ Petal.Length: num  1.4 1.4 1.3 1.5 1.4 1.7 1.4 1.5 1.4 1.5 ...
##  $ Petal.Width : num  0.2 0.2 0.2 0.2 0.2 0.4 0.3 0.2 0.2 0.1 ...
##  $ Species     : Factor w/ 3 levels "setosa","versicolor",..: 1 1 1 1 1 1 1 1 1 1 ...
head(iris, 5)
##   Sepal.Length Sepal.Width Petal.Length Petal.Width Species
## 1          5.1         3.5          1.4         0.2  setosa
## 2          4.9         3.0          1.4         0.2  setosa
## 3          4.7         3.2          1.3         0.2  setosa
## 4          4.6         3.1          1.5         0.2  setosa
## 5          5.0         3.6          1.4         0.2  setosa

This dataset has 3 different species of irises, with 150 observations of measurements of petals and sepals. Let’s try to cluster based on sepal length. Since we know that there are 3 different species in the dataset, we can assume that there will be three clusters.

To start, our data looks like this:

library(ggplot2)
ggplot(iris, aes(Petal.Length, Petal.Width)) + geom_point()

Let’s use the kmeans function.

iris_cluster <- kmeans(iris[, 3:4], 3)
iris_cluster
## K-means clustering with 3 clusters of sizes 50, 54, 46
## 
## Cluster means:
##   Petal.Length Petal.Width
## 1     1.462000    0.246000
## 2     4.292593    1.359259
## 3     5.626087    2.047826
## 
## Clustering vector:
##   [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
##  [36] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
##  [71] 2 2 2 2 2 2 2 3 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3
## [106] 3 2 3 3 3 3 3 3 3 3 3 3 3 3 2 3 3 3 2 3 3 2 2 3 3 3 3 3 3 3 3 3 3 2 3
## [141] 3 3 3 3 3 3 3 3 3 3
## 
## Within cluster sum of squares by cluster:
## [1]  2.02200 14.22741 15.16348
##  (between_SS / total_SS =  94.3 %)
## 
## Available components:
## 
## [1] "cluster"      "centers"      "totss"        "withinss"    
## [5] "tot.withinss" "betweenss"    "size"         "iter"        
## [9] "ifault"

If we plot the graph with colors for each cluster, we get the following:

ggplot(iris, aes(Petal.Length, Petal.Width, color = as.factor(iris_cluster$cluster))) + geom_point()

Let’s compare this to actual species given.

iris$Species.Sort <- factor(iris$Species, levels=c("virginica", "versicolor", "setosa"))
ggplot(iris, aes(Petal.Length, Petal.Width, color = factor(iris$Species.Sort))) + geom_point()

There are some minor differences. We can see where kmeans failed.

table(iris_cluster$cluster, iris$Species)
##    
##     setosa versicolor virginica
##   1     50          0         0
##   2      0         48         6
##   3      0          2        44

Different Options

R’s built-in kmeans also has some helpful initialization options:

  • iter.max: number of iterations to run. Default is set to 10.
    • In general, more is better, but K-Means converges so quickly that a large number usually isn’t needed.
  • nstart: number of times to try to pick the best center. Default is set to 1.
  • algorithm: default is Lloyd/Forgy (described above). MacQueen and Hartigan-Wong have essentially the same distance calculation, but are smarter performance-wise. For instance, MacQueen updates the centroids any time a point is moved.

Drawbacks

K-Means does have a few drawbacks.

  • Number of clusters must be set a priori.
  • Variables must be continuous (due to calculation of distance).
  • Algorithm converges to a local but not necessarily global optimum.
  • Each point must be assigned to only one cluster.
  • Heavily affected by outliers.
  • Inaccurate for concave datasets.

The last three points are particularly important to note. In these cases, K-Means may not be the best clustering algorithm of choice. We can see this in the next section.

Concave Dataset Example

Let’s take a dataset that has a concave U shape in data. For this, we can use the artificially dataset smiley.

library(mlbench)
smiley <- mlbench.smiley()
str(smiley)
## List of 2
##  $ x      : num [1:500, 1:2] -0.802 -0.743 -0.933 -0.924 -0.686 ...
##   ..- attr(*, "dimnames")=List of 2
##   .. ..$ : NULL
##   .. ..$ : chr [1:2] "x4" ""
##  $ classes: Factor w/ 4 levels "1","2","3","4": 1 1 1 1 1 1 1 1 1 1 ...
##  - attr(*, "class")= chr [1:2] "mlbench.smiley" "mlbench"
smiley_data <- data.frame(smiley$x)
colnames(smiley_data)[colnames(smiley_data)=="x4"] <- "x"
colnames(smiley_data)[colnames(smiley_data)=="V2"] <- "y"
str(smiley_data)
## 'data.frame':    500 obs. of  2 variables:
##  $ x: num  -0.802 -0.743 -0.933 -0.924 -0.686 ...
##  $ y: num  1.035 1.01 0.965 0.939 0.926 ...
ggplot(smiley_data, aes(x, y)) + geom_point()

We know there’s four clusters (two eyes, a nose, and a mouth). Let’s take a look at what kmeans produces.

smiley_cluster <- kmeans(smiley_data, 4, iter.max = 50)
smiley_cluster
## K-means clustering with 4 clusters of sizes 80, 129, 208, 83
## 
## Cluster means:
##            x          y
## 1 -0.6991541 -0.4581949
## 2  0.4137118 -0.7324549
## 3 -0.3127123  0.6038361
## 4  0.8117034  1.0022588
## 
## Clustering vector:
##   [1] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
##  [36] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
##  [71] 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
## [106] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
## [141] 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3
## [176] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## [211] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## [246] 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
## [281] 3 3 3 3 3 3 3 3 3 3 3 2 1 2 1 2 1 1 2 2 2 2 2 1 2 2 2 2 1 2 2 2 2 2 2
## [316] 1 2 2 1 1 1 1 2 1 2 2 2 2 2 2 2 2 1 1 2 1 1 1 2 1 2 1 1 1 2 1 1 2 1 1
## [351] 2 2 2 2 2 2 1 2 2 2 2 1 2 2 2 2 2 2 2 2 2 1 2 1 1 2 2 1 1 2 1 2 1 2 1
## [386] 2 1 2 2 2 2 1 2 1 1 1 2 1 2 2 2 2 1 1 1 2 2 1 1 1 1 1 2 1 1 2 2 2 2 2
## [421] 2 2 2 1 2 2 2 1 2 2 1 2 2 2 2 2 1 2 1 2 1 1 1 2 2 2 1 1 2 1 2 1 1 1 1
## [456] 2 2 2 1 2 2 1 2 2 2 2 1 1 2 2 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 1 2 2 1 2
## [491] 1 2 1 2 1 1 2 2 2 1
## 
## Within cluster sum of squares by cluster:
## [1] 11.10700 24.52663 59.15788  1.76754
##  (between_SS / total_SS =  79.3 %)
## 
## Available components:
## 
## [1] "cluster"      "centers"      "totss"        "withinss"    
## [5] "tot.withinss" "betweenss"    "size"         "iter"        
## [9] "ifault"
smiley_cluster$cluster_factor <- as.factor(smiley_cluster$cluster)
centroids <- data.frame(smiley_cluster$centers)
centroids
##            x          y
## 1 -0.6991541 -0.4581949
## 2  0.4137118 -0.7324549
## 3 -0.3127123  0.6038361
## 4  0.8117034  1.0022588
ggplot(data=smiley_data, aes(x, y, color=smiley_cluster$cluster_factor)) + geom_point() + geom_point(data=centroids, aes(x, y), color="black", size=3) 

We mark the centroids for each cluster. As we can see, kmeans struggled a lot with the U shape of the mouth, breaking it up into two curves. In addition, with the way the clusters are defined, it grouped the left eye with the nose.

A better algorithm for clustering in these instances is a mixture model, which uses a probability distribution as opposed to hard assignments. The Gaussian mixture model in particular handles data with more skew/kurtosis and is more flexible in terms of cluster covariance, which means it can handle shapes that K-Means cannot.

We can take a look at how GMM handles this dataset.

We initialize GMM the same way we initialize kmeans, giving it the number of clusters we want.

library(mclust)
## Package 'mclust' version 5.3
## Type 'citation("mclust")' for citing this R package in publications.
smiley_gmm <- Mclust(smiley_data, G=4)
smiley_gmm
## 'Mclust' model object:
##  best model: diagonal, varying volume and shape (VVI) with 4 components
ggplot(data=smiley_data, aes(x, y, color=as.factor(smiley_gmm$classification))) + geom_point()

Here, we can see that even an out of the box GMM call gives us much more accurate clusters.

FOAD ———————————————————————-

Overview

General idea behind clustering analysis is to predict or discover outcomes from measured predictors.The goal of this overview was to try many methods on a dataset often used for cluster analysis. Provided are some visualizations between methods of dendrogram plotting, as well as an example utility of hClust to order variables in a correlation matrix.

Rank observations by Area

seed.df2 <- transform(seed.df, 
          RankArea = rank(-Area, 
                           ties.method = "average"))
seed.df <- seed.df2[,-8]
seed_labels <- seed.df2[,8]
require(colorspace)
seed_col <- sort(sequential_hcl(203, power = 2))[as.numeric(seed_labels)] #Sequential HCL with quadratic power
rm(seed_labels)
rm(seed.df2)

Setup pairwise scatterplots of all dataset variables

#Plot matrix
pairs(seed.df, col = seed_col,
      main = "Pairwise comparison of seeds parameters ranked by area",
      lower.panel = NULL,
      pch = 20, cex = 1.0)

Principle component analysis with ‘pam’ around-medoids on fixed clusters

Clustering method comparison from “hclust”

Determine number of clusters using groups within sums of squares (WSS)

Correlation matrix setup of seed parameters

hClust of the correlation matrix

Dendrogram of the clustered correlation matrix

MALAVIKA ——————————————————————

Try out the clustering models

I have used a data set from a fine jewelry company. I am trying to find if there are any customer segments that can be made based on states. I am using the gemestones and average price as variables to group states into clusters and get some insight on which states tend to have similar buying patterns in terms of gemstones and average price.

data<-read.csv("Files/Malavika/cluster.csv",header = TRUE, sep=",")
str(data)
## 'data.frame':    53 obs. of  23 variables:
##  $ State          : Factor w/ 53 levels "Alabama","Alaska",..: 1 2 3 4 5 6 7 8 9 10 ...
##  $ Amethyst       : num  619 601 NA 545 342 ...
##  $ Aquamarine     : num  1413 1193 NA 865 NA ...
##  $ Black.Diamond  : num  NA NA NA 923 NA ...
##  $ Blue.Topaz     : num  936 NA NA 552 1075 ...
##  $ Citrine        : num  424 NA NA 254 581 ...
##  $ Diamond        : num  NA NA NA NA NA ...
##  $ Emerald        : num  1564 1404 NA 1836 1834 ...
##  $ Garnet         : num  930 NA NA 1352 NA ...
##  $ Morganite      : num  NA NA NA 696 NA NA 641 NA NA NA ...
##  $ Peridot        : num  549 NA NA 968 NA ...
##  $ Pink.Sapphire  : num  NA 2393 NA 736 NA ...
##  $ Pink.Tourmaline: num  NA 1000 1369 1750 NA ...
##  $ Plain          : num  NA NA NA 536 NA ...
##  $ Ruby           : num  1991 NA NA 2211 1822 ...
##  $ Sapphire       : num  1368 2343 NA 1325 NA ...
##  $ SI.Diamond     : num  NA 1799 NA 1325 577 ...
##  $ Tanzanite      : num  280 1453 NA 1298 680 ...
##  $ VS.Diamond     : num  NA NA NA 1169 NA ...
##  $ VVS.Diamond    : num  NA NA NA NA NA ...
##  $ White.Sapphire : num  NA NA NA 673 NA NA 880 NA NA 460 ...
##  $ White.Topaz    : num  NA NA NA NA NA ...
##  $ Grand.Total    : num  1105 1292 1369 1193 780 ...
data[is.na(data)]<-0
datascaled<-scale(data[2:22])
datanew<-data.frame(data$State,datascaled)
rownames(datanew) <- data$State
dm<-dist(datanew,method="euclidean")
## Warning in dist(datanew, method = "euclidean"): NAs introduced by coercion
#using the euclidean method of calculating distance.
as.matrix(dm)
##                 Alabama   Alaska  Alberta  Arizona Arkansas Armed Forces
## Alabama        0.000000 6.458014 5.938607 5.549891 4.225919     5.451675
## Alaska         6.458014 0.000000 6.556102 6.622259 6.829761     6.877432
## Alberta        5.938607 6.556102 0.000000 7.222525 4.561552     4.434178
## Arizona        5.549891 6.622259 7.222525 0.000000 6.735130     6.715123
## Arkansas       4.225919 6.829761 4.561552 6.735130 0.000000     3.856893
## Armed Forces   5.451675 6.877432 4.434178 6.715123 3.856893     0.000000
## California     7.185794 6.907958 8.681089 4.875951 7.785277     8.091055
## Colorado       6.591717 6.209829 8.190231 5.910030 7.479995     7.642559
## Connecticut    4.785040 5.595497 5.960058 5.695072 5.226764     5.140044
## Delaware       7.237935 6.364649 7.453248 7.718397 6.896935     6.053158
## Florida        5.885729 7.576403 8.162022 4.887118 6.742404     7.057881
## Georgia        4.512216 6.732959 5.511730 4.480849 5.650880     4.989234
## Hawaii         6.569188 7.208045 7.783550 7.032423 7.081671     7.138056
## Idaho          5.102335 5.094759 6.133954 6.051524 5.775522     5.548175
## Illinois       6.314432 8.150782 8.884725 6.834064 7.435443     7.819200
## Indiana        4.983077 5.753462 6.169430 4.805129 5.231410     5.089304
## Iowa           6.299427 7.268197 6.298100 5.563254 5.950419     4.851111
## Kansas         7.397043 8.604726 7.411567 7.720489 7.376899     7.013929
## Kentucky       4.372414 5.769216 5.995290 6.288914 5.516032     5.595574
## Louisiana      4.452699 6.615511 7.069131 5.156843 5.119941     5.887849
## Maine          5.293218 5.282709 4.221968 5.877516 4.867506     5.569170
## Maryland       8.093639 8.003989 9.753571 6.472942 8.745200     8.728376
## Massachusetts  5.392543 6.375656 7.706540 3.472857 6.839165     6.466005
## Michigan       4.808472 7.012517 6.792158 5.303960 5.515546     6.748488
## Minnesota      4.700909 7.964598 7.831709 5.807552 4.853899     6.301982
## Mississippi    3.774816 5.326722 4.904107 6.726372 4.701935     5.189008
## Missouri       3.563939 6.745481 7.075855 5.088364 4.981031     5.780591
## Montana        4.866612 4.608081 4.490560 6.773789 4.515885     4.084661
## Nebraska       6.504862 6.970077 6.503546 5.786813 7.125340     7.069650
## Nevada         5.267194 7.276109 5.753505 6.652513 4.865336     4.800257
## New Hampshire  4.030752 6.240840 4.718106 6.225590 3.270077     4.202395
## New Jersey     6.503568 7.378009 6.662825 4.259374 6.587982     6.468605
## New Mexico     5.618718 6.719694 5.653526 7.184587 5.206213     5.551038
## New York       6.249322 8.427771 8.855705 5.063408 7.603144     7.606226
## North Carolina 6.998369 7.753897 8.426676 6.006001 7.790330     7.902163
## North Dakota   9.191299 8.823546 8.950400 9.631094 7.682169     8.965781
## Ohio           4.773265 6.663065 6.491194 3.641898 5.862354     5.823620
## Oklahoma       4.782860 6.996699 6.270884 4.431446 6.146751     5.576427
## Oregon         4.064872 6.727814 6.662351 6.005467 3.975158     5.316895
## Pennsylvania   7.450679 7.121428 8.635543 5.977805 8.198800     8.179681
## Rhode Island   4.232285 7.082969 6.718567 7.192322 5.380866     5.610010
## South Carolina 6.555781 8.974856 9.576543 5.262589 7.755976     8.262035
## South Dakota   5.319670 6.607146 2.393359 7.663449 3.857254     3.682592
## Tennessee      4.606873 7.524248 8.198124 6.013190 5.811730     6.814115
## Texas          6.197279 7.510870 8.098935 5.184887 6.629659     7.168956
## Utah           8.762303 9.297339 9.084360 8.704028 8.516727     7.466403
## Vermont        5.149541 6.761137 2.498740 7.621458 3.320485     3.941580
## Virginia       4.944948 6.504468 6.219916 3.627328 6.087228     5.632139
## Washington     5.716118 7.197493 7.648839 6.318643 5.974328     5.627650
## West Virginia  4.407156 7.237112 4.366555 6.781851 4.848998     4.866620
## Wisconsin      6.120516 6.813791 7.019624 5.826823 5.454567     5.995199
## Wyoming        4.915825 6.411280 6.064700 7.182683 5.037459     4.899935
## Grand Total    6.218044 6.649416 7.984547 4.274130 6.983158     7.271430
##                California  Colorado Connecticut Delaware   Florida
## Alabama          7.185794  6.591717    4.785040 7.237935  5.885729
## Alaska           6.907958  6.209829    5.595497 6.364649  7.576403
## Alberta          8.681089  8.190231    5.960058 7.453248  8.162022
## Arizona          4.875951  5.910030    5.695072 7.718397  4.887118
## Arkansas         7.785277  7.479995    5.226764 6.896935  6.742404
## Armed Forces     8.091055  7.642559    5.140044 6.053158  7.057881
## California       0.000000  5.424481    6.369320 7.979402  4.452789
## Colorado         5.424481  0.000000    5.362504 8.141139  5.915216
## Connecticut      6.369320  5.362504    0.000000 6.585539  5.389453
## Delaware         7.979402  8.141139    6.585539 0.000000  7.176644
## Florida          4.452789  5.915216    5.389453 7.176644  0.000000
## Georgia          6.092672  5.523206    5.228045 7.529675  5.337328
## Hawaii           7.630064  6.782499    6.197700 9.302851  8.393797
## Idaho            7.102374  6.573462    5.701367 6.367252  6.208132
## Illinois         5.449606  7.188713    7.302879 7.746595  5.961508
## Indiana          5.797307  5.889190    4.414236 5.741255  4.019368
## Iowa             6.204609  6.715460    4.958656 6.599537  4.921421
## Kansas           8.905964  7.312476    6.746150 8.363158  7.343258
## Kentucky         7.089404  6.464372    5.170206 6.313514  6.053618
## Louisiana        6.625641  6.801513    4.931743 7.081129  4.463888
## Maine            7.542683  7.014015    6.139437 7.340093  7.105972
## Maryland         4.789696  7.508220    8.533523 8.778590  6.973649
## Massachusetts    5.114866  6.256095    6.015409 7.616973  5.167417
## Michigan         6.374332  6.872394    5.432203 7.383262  4.849294
## Minnesota        6.682124  7.076054    6.200400 8.318038  4.977421
## Mississippi      7.925699  7.382828    5.060682 6.392635  6.655417
## Missouri         6.526190  6.784031    5.610006 7.697947  4.952425
## Montana          7.809604  7.077124    4.990817 4.103663  6.685003
## Nebraska         7.430779  7.170333    7.197735 8.554941  6.940012
## Nevada           7.519307  7.502369    6.303371 7.145776  6.509763
## New Hampshire    7.590397  7.041917    4.371487 6.281477  6.315661
## New Jersey       4.796961  6.661809    5.706699 7.402078  3.897598
## New Mexico       7.457747  5.134640    5.859649 7.124186  6.833203
## New York         4.772579  6.858847    6.406783 8.223225  4.376549
## North Carolina   6.486866  6.829243    6.017827 8.422485  4.232857
## North Dakota    10.648191 10.732955    9.436172 9.572150 10.360092
## Ohio             5.627328  5.889494    5.277582 7.404879  3.911594
## Oklahoma         6.310038  6.628926    6.126956 6.920917  5.626614
## Oregon           6.624895  5.970400    4.312459 7.607843  5.303754
## Pennsylvania     6.125095  4.381066    6.960579 8.056631  6.424759
## Rhode Island     7.723668  6.896976    5.748315 7.910543  5.948382
## South Carolina   7.205556  8.308857    7.034052 9.339922  5.933449
## South Dakota     8.792791  8.120123    5.658362 6.884977  7.843544
## Tennessee        6.406763  6.874478    5.669693 7.565404  4.748354
## Texas            5.388334  4.889869    6.006434 7.884593  4.433368
## Utah             8.061904  8.058331    6.676023 8.382922  6.517269
## Vermont          8.708832  8.042605    5.436103 7.148112  7.686588
## Virginia         5.707211  5.582686    5.402257 6.793980  4.799523
## Washington       6.853838  6.057213    5.590082 6.302715  4.921293
## West Virginia    8.845652  8.273067    5.056785 7.814470  7.831506
## Wisconsin        6.370484  7.417557    6.862335 7.082930  5.833695
## Wyoming          7.770176  7.430232    5.426891 6.325067  6.679687
## Grand Total      2.442488  4.109117    5.414111 7.635680  3.667190
##                 Georgia    Hawaii    Idaho  Illinois  Indiana     Iowa
## Alabama        4.512216  6.569188 5.102335  6.314432 4.983077 6.299427
## Alaska         6.732959  7.208045 5.094759  8.150782 5.753462 7.268197
## Alberta        5.511730  7.783550 6.133954  8.884725 6.169430 6.298100
## Arizona        4.480849  7.032423 6.051524  6.834064 4.805129 5.563254
## Arkansas       5.650880  7.081671 5.775522  7.435443 5.231410 5.950419
## Armed Forces   4.989234  7.138056 5.548175  7.819200 5.089304 4.851111
## California     6.092672  7.630064 7.102374  5.449606 5.797307 6.204609
## Colorado       5.523206  6.782499 6.573462  7.188713 5.889190 6.715460
## Connecticut    5.228045  6.197700 5.701367  7.302879 4.414236 4.958656
## Delaware       7.529675  9.302851 6.367252  7.746595 5.741255 6.599537
## Florida        5.337328  8.393797 6.208132  5.961508 4.019368 4.921421
## Georgia        0.000000  6.750297 5.281656  6.711226 4.271838 5.203500
## Hawaii         6.750297  0.000000 6.888715  8.628018 7.708991 7.140632
## Idaho          5.281656  6.888715 0.000000  6.904785 4.867216 5.097081
## Illinois       6.711226  8.628018 6.904785  0.000000 6.216753 6.914213
## Indiana        4.271838  7.708991 4.867216  6.216753 0.000000 4.523287
## Iowa           5.203500  7.140632 5.097081  6.914213 4.523287 0.000000
## Kansas         7.202238  8.267355 7.136644  9.203973 6.740914 7.210568
## Kentucky       4.979064  6.552144 2.494074  6.998819 4.599057 4.924156
## Louisiana      5.630785  7.895766 6.203301  6.415804 4.390920 6.064292
## Maine          5.195503  7.091681 4.271554  7.492483 5.311778 6.558231
## Maryland       7.767793  9.256796 8.109737  5.487027 7.611905 8.470258
## Massachusetts  4.452077  7.356625 5.613256  7.261024 5.293372 6.224197
## Michigan       5.103176  8.929174 6.549170  7.689075 5.064910 7.113830
## Minnesota      5.827999  8.418906 6.214278  6.565767 5.137562 6.192119
## Mississippi    5.426846  8.156091 4.866298  7.037977 4.910308 6.548849
## Missouri       5.159653  7.046491 5.540222  6.240232 4.926120 6.275760
## Montana        5.560399  7.374638 3.788530  7.343585 4.461821 5.461255
## Nebraska       4.877631  7.583563 5.851732  8.863687 6.233945 6.981872
## Nevada         5.539872  7.003364 6.616295  7.567007 5.426482 6.349178
## New Hampshire  5.702455  6.446677 4.848501  7.203726 4.623185 4.717626
## New Jersey     4.481658  8.812251 6.854996  7.084215 4.269454 5.312699
## New Mexico     5.090302  7.132172 6.212914  7.347758 6.066660 6.372687
## New York       5.592145  9.094107 6.779256  5.086406 5.189313 6.267474
## North Carolina 6.492492  8.674060 7.446499  8.295646 5.182880 6.843864
## North Dakota   9.624348 10.192808 8.238749 10.458669 8.565928 9.518730
## Ohio           3.595545  7.206117 5.227263  6.565286 4.250188 4.953120
## Oklahoma       4.192569  5.888685 4.916438  6.813031 5.396766 4.960254
## Oregon         5.470112  6.533233 6.189566  7.438703 5.066064 5.964436
## Pennsylvania   6.502955  8.631923 7.171559  7.700079 6.071833 7.524562
## Rhode Island   4.828714  8.186827 5.970258  7.787660 5.838866 6.901866
## South Carolina 7.392903  8.869245 8.173157  7.520300 6.086222 7.469876
## South Dakota   5.734253  7.479454 5.648095  8.352783 5.972264 6.212878
## Tennessee      5.705411  7.902641 5.868657  6.466157 5.270415 6.485607
## Texas          5.219959  7.829138 6.090771  6.668738 5.543089 6.001264
## Utah           7.053998  9.399234 7.865078  8.723810 5.477808 6.477314
## Vermont        5.819848  7.444364 5.642209  8.461158 6.115764 6.105436
## Virginia       3.542946  6.734386 4.879180  7.167333 4.559654 4.543963
## Washington     5.142458  8.710108 6.069777  7.090325 3.830936 6.664780
## West Virginia  5.634109  7.711825 6.437182  8.037606 6.165578 6.860840
## Wisconsin      6.279280  7.694510 5.308922  6.725349 5.284446 6.245025
## Wyoming        5.185658  7.762876 6.195199  6.934846 4.578402 6.896238
## Grand Total    5.089513  7.333753 6.373841  5.266276 4.921929 5.891074
##                   Kansas Kentucky Louisiana    Maine  Maryland
## Alabama         7.397043 4.372414  4.452699 5.293218  8.093639
## Alaska          8.604726 5.769216  6.615511 5.282709  8.003989
## Alberta         7.411567 5.995290  7.069131 4.221968  9.753571
## Arizona         7.720489 6.288914  5.156843 5.877516  6.472942
## Arkansas        7.376899 5.516032  5.119941 4.867506  8.745200
## Armed Forces    7.013929 5.595574  5.887849 5.569170  8.728376
## California      8.905964 7.089404  6.625641 7.542683  4.789696
## Colorado        7.312476 6.464372  6.801513 7.014015  7.508220
## Connecticut     6.746150 5.170206  4.931743 6.139437  8.533523
## Delaware        8.363158 6.313514  7.081129 7.340093  8.778590
## Florida         7.343258 6.053618  4.463888 7.105972  6.973649
## Georgia         7.202238 4.979064  5.630785 5.195503  7.767793
## Hawaii          8.267355 6.552144  7.895766 7.091681  9.256796
## Idaho           7.136644 2.494074  6.203301 4.271554  8.109737
## Illinois        9.203973 6.998819  6.415804 7.492483  5.487027
## Indiana         6.740914 4.599057  4.390920 5.311778  7.611905
## Iowa            7.210568 4.924156  6.064292 6.558231  8.470258
## Kansas          0.000000 6.970763  8.089403 6.351712  9.807778
## Kentucky        6.970763 0.000000  6.548283 5.197295  8.741115
## Louisiana       8.089403 6.548283  0.000000 6.087485  7.366151
## Maine           6.351712 5.197295  6.087485 0.000000  8.025654
## Maryland        9.807778 8.741115  7.366151 8.025654  0.000000
## Massachusetts   8.489479 5.846212  5.674934 6.479105  6.228201
## Michigan        8.429864 6.243535  4.637338 6.352609  8.002383
## Minnesota       8.614830 6.214171  4.670789 6.945337  8.044596
## Mississippi     7.668144 4.973478  5.074344 4.386038  8.444188
## Missouri        7.917208 5.903365  3.131845 5.631463  6.981011
## Montana         7.038329 3.873661  5.549094 4.434018  8.647836
## Nebraska        8.187371 5.896632  7.040334 5.971019  9.279251
## Nevada          6.935151 6.315286  5.301318 5.710532  7.452606
## New Hampshire   6.802157 4.147641  4.948457 5.206466  8.835816
## New Jersey      7.941635 6.791874  5.549812 6.483843  7.403636
## New Mexico      7.490999 6.031380  6.159567 5.789630  8.474956
## New York        8.285018 6.533630  6.272072 7.698165  5.853669
## North Carolina  7.521103 7.470538  5.516528 7.415345  8.744170
## North Dakota   10.389917 8.678948  9.416963 7.148142 11.242309
## Ohio            7.221690 5.449320  3.804193 5.393729  6.972090
## Oklahoma        6.897179 4.694252  5.899778 5.448801  7.121522
## Oregon          7.154538 5.628527  4.420353 6.535852  8.383445
## Pennsylvania    8.320207 7.199063  7.156796 7.458868  7.034796
## Rhode Island    7.834624 5.756226  5.284769 6.535732  8.722184
## South Carolina  9.817396 8.061831  5.131123 8.656070  8.216247
## South Dakota    6.947364 5.475771  6.538900 4.413409  9.439669
## Tennessee       8.943009 5.576714  4.917737 7.324797  8.450909
## Texas           8.209651 6.420524  5.131024 6.607062  6.665696
## Utah            8.623943 7.300432  8.260537 8.981313 10.223566
## Vermont         7.076865 5.384558  6.440858 4.526146  9.669427
## Virginia        6.050832 4.688628  5.914366 5.310607  7.594713
## Washington      6.329396 5.919382  5.562652 6.613086  7.850963
## West Virginia   7.711189 6.070705  5.818086 5.441244  9.255390
## Wisconsin       7.953246 6.097142  5.311139 5.189702  5.931201
## Wyoming         8.005924 6.064219  5.393941 5.563723  8.299394
## Grand Total     7.772135 6.443837  5.507967 6.691132  5.075635
##                Massachusetts Michigan Minnesota Mississippi Missouri
## Alabama             5.392543 4.808472  4.700909    3.774816 3.563939
## Alaska              6.375656 7.012517  7.964598    5.326722 6.745481
## Alberta             7.706540 6.792158  7.831709    4.904107 7.075855
## Arizona             3.472857 5.303960  5.807552    6.726372 5.088364
## Arkansas            6.839165 5.515546  4.853899    4.701935 4.981031
## Armed Forces        6.466005 6.748488  6.301982    5.189008 5.780591
## California          5.114866 6.374332  6.682124    7.925699 6.526190
## Colorado            6.256095 6.872394  7.076054    7.382828 6.784031
## Connecticut         6.015409 5.432203  6.200400    5.060682 5.610006
## Delaware            7.616973 7.383262  8.318038    6.392635 7.697947
## Florida             5.167417 4.849294  4.977421    6.655417 4.952425
## Georgia             4.452077 5.103176  5.827999    5.426846 5.159653
## Hawaii              7.356625 8.929174  8.418906    8.156091 7.046491
## Idaho               5.613256 6.549170  6.214278    4.866298 5.540222
## Illinois            7.261024 7.689075  6.565767    7.037977 6.240232
## Indiana             5.293372 5.064910  5.137562    4.910308 4.926120
## Iowa                6.224197 7.113830  6.192119    6.548849 6.275760
## Kansas              8.489479 8.429864  8.614830    7.668144 7.917208
## Kentucky            5.846212 6.243535  6.214171    4.973478 5.903365
## Louisiana           5.674934 4.637338  4.670789    5.074344 3.131845
## Maine               6.479105 6.352609  6.945337    4.386038 5.631463
## Maryland            6.228201 8.002383  8.044596    8.444188 6.981011
## Massachusetts       0.000000 4.825644  5.631808    6.289814 4.805791
## Michigan            4.825644 0.000000  5.121099    5.563666 5.164710
## Minnesota           5.631808 5.121099  0.000000    6.196047 3.969467
## Mississippi         6.289814 5.563666  6.196047    0.000000 4.812340
## Missouri            4.805791 5.164710  3.969467    4.812340 0.000000
## Montana             6.635010 6.144631  6.848451    3.690420 5.865714
## Nebraska            6.090249 6.008477  7.449317    7.686031 6.837371
## Nevada              6.679064 6.719875  7.032359    5.762934 5.173519
## New Hampshire       6.841214 6.035567  5.663739    4.955200 5.299715
## New Jersey          4.888535 4.406335  5.979373    6.664380 5.827660
## New Mexico          7.163839 6.585197  7.041983    5.618244 6.438727
## New York            5.282642 5.783508  5.744175    7.451404 6.328482
## North Carolina      6.159327 6.011770  6.703591    7.009354 5.452462
## North Dakota        9.636062 9.644444  9.959241    8.182110 9.554006
## Ohio                4.045568 4.501246  5.488255    5.623936 4.521139
## Oklahoma            4.742192 6.480878  6.672593    6.017652 5.189413
## Oregon              5.804385 4.784171  4.123454    5.999159 4.395396
## Pennsylvania        6.115796 6.939652  7.702952    7.743356 6.901824
## Rhode Island        5.941597 5.019436  5.414325    5.104582 4.639582
## South Carolina      6.024039 6.590600  5.629351    7.967452 5.711664
## South Dakota        7.656587 6.814901  7.328437    4.131344 6.529536
## Tennessee           5.321992 4.473132  3.623236    6.327798 4.268819
## Texas               5.386818 5.324044  6.008542    7.061935 5.292006
## Utah                8.659978 8.576313  8.438402    8.579812 8.560707
## Vermont             7.614492 6.396823  6.947430    4.289154 6.470481
## Virginia            4.246912 5.554418  6.001199    6.106232 5.128780
## Washington          5.675188 5.106184  5.338240    6.103148 5.563479
## West Virginia       7.029111 6.093132  7.228503    4.219397 6.275096
## Wisconsin           5.924734 6.477349  5.868447    6.167304 4.845933
## Wyoming             6.739440 6.319361  6.656422    3.849856 5.044458
## Grand Total         4.526745 5.275256  5.802195    7.003535 5.443779
##                 Montana  Nebraska   Nevada New Hampshire New Jersey
## Alabama        4.866612  6.504862 5.267194      4.030752   6.503568
## Alaska         4.608081  6.970077 7.276109      6.240840   7.378009
## Alberta        4.490560  6.503546 5.753505      4.718106   6.662825
## Arizona        6.773789  5.786813 6.652513      6.225590   4.259374
## Arkansas       4.515885  7.125340 4.865336      3.270077   6.587982
## Armed Forces   4.084661  7.069650 4.800257      4.202395   6.468605
## California     7.809604  7.430779 7.519307      7.590397   4.796961
## Colorado       7.077124  7.170333 7.502369      7.041917   6.661809
## Connecticut    4.990817  7.197735 6.303371      4.371487   5.706699
## Delaware       4.103663  8.554941 7.145776      6.281477   7.402078
## Florida        6.685003  6.940012 6.509763      6.315661   3.897598
## Georgia        5.560399  4.877631 5.539872      5.702455   4.481658
## Hawaii         7.374638  7.583563 7.003364      6.446677   8.812251
## Idaho          3.788530  5.851732 6.616295      4.848501   6.854996
## Illinois       7.343585  8.863687 7.567007      7.203726   7.084215
## Indiana        4.461821  6.233945 5.426482      4.623185   4.269454
## Iowa           5.461255  6.981872 6.349178      4.717626   5.312699
## Kansas         7.038329  8.187371 6.935151      6.802157   7.941635
## Kentucky       3.873661  5.896632 6.315286      4.147641   6.791874
## Louisiana      5.549094  7.040334 5.301318      4.948457   5.549812
## Maine          4.434018  5.971019 5.710532      5.206466   6.483843
## Maryland       8.647836  9.279251 7.452606      8.835816   7.403636
## Massachusetts  6.635010  6.090249 6.679064      6.841214   4.888535
## Michigan       6.144631  6.008477 6.719875      6.035567   4.406335
## Minnesota      6.848451  7.449317 7.032359      5.663739   5.979373
## Mississippi    3.690420  7.686031 5.762934      4.955200   6.664380
## Missouri       5.865714  6.837371 5.173519      5.299715   5.827660
## Montana        0.000000  6.457812 5.214852      3.654956   6.798800
## Nebraska       6.457812  0.000000 7.274766      6.379948   5.897104
## Nevada         5.214852  7.274766 0.000000      4.822413   6.894747
## New Hampshire  3.654956  6.379948 4.822413      0.000000   6.673016
## New Jersey     6.798800  5.897104 6.894747      6.673016   0.000000
## New Mexico     5.065790  7.149134 5.371428      5.489742   7.156392
## New York       7.773660  7.795013 7.580724      7.227466   5.276604
## North Carolina 7.566820  7.520702 7.471178      7.531054   4.780133
## North Dakota   8.186247 10.678236 9.473165      8.544177  10.036331
## Ohio           5.622588  5.277986 5.213383      5.376355   4.490014
## Oklahoma       5.354831  6.000114 4.569980      5.306841   6.171824
## Oregon         5.768759  6.209362 5.556000      4.036584   6.037221
## Pennsylvania   7.505236  7.583019 7.676239      7.633546   6.400008
## Rhode Island   5.712947  6.402327 5.756628      5.879335   6.538849
## South Carolina 8.476217  8.596137 8.074329      7.123741   6.770492
## South Dakota   3.676748  7.282063 4.889343      4.197512   7.322902
## Tennessee      6.513385  6.486458 7.679118      5.918545   5.684491
## Texas          6.890048  6.905791 6.450885      6.633053   5.511662
## Utah           7.840617  9.454008 8.107921      8.117123   7.277510
## Vermont        3.963550  7.125633 5.330111      3.906106   7.174625
## Virginia       5.604587  4.853765 5.857743      5.438581   4.429281
## Washington     5.735890  7.012228 6.296989      6.319814   5.592976
## West Virginia  5.100971  7.654238 5.875271      4.562432   7.293378
## Wisconsin      5.675933  7.611519 4.729871      5.810459   6.648407
## Wyoming        4.719607  8.279431 4.955522      5.926614   6.706337
## Grand Total    7.067734  6.657441 6.891992      6.812184   4.143404
##                New Mexico  New York North Carolina North Dakota     Ohio
## Alabama          5.618718  6.249322       6.998369     9.191299 4.773265
## Alaska           6.719694  8.427771       7.753897     8.823546 6.663065
## Alberta          5.653526  8.855705       8.426676     8.950400 6.491194
## Arizona          7.184587  5.063408       6.006001     9.631094 3.641898
## Arkansas         5.206213  7.603144       7.790330     7.682169 5.862354
## Armed Forces     5.551038  7.606226       7.902163     8.965781 5.823620
## California       7.457747  4.772579       6.486866    10.648191 5.627328
## Colorado         5.134640  6.858847       6.829243    10.732955 5.889494
## Connecticut      5.859649  6.406783       6.017827     9.436172 5.277582
## Delaware         7.124186  8.223225       8.422485     9.572150 7.404879
## Florida          6.833203  4.376549       4.232857    10.360092 3.911594
## Georgia          5.090302  5.592145       6.492492     9.624348 3.595545
## Hawaii           7.132172  9.094107       8.674060    10.192808 7.206117
## Idaho            6.212914  6.779256       7.446499     8.238749 5.227263
## Illinois         7.347758  5.086406       8.295646    10.458669 6.565286
## Indiana          6.066660  5.189313       5.182880     8.565928 4.250188
## Iowa             6.372687  6.267474       6.843864     9.518730 4.953120
## Kansas           7.490999  8.285018       7.521103    10.389917 7.221690
## Kentucky         6.031380  6.533630       7.470538     8.678948 5.449320
## Louisiana        6.159567  6.272072       5.516528     9.416963 3.804193
## Maine            5.789630  7.698165       7.415345     7.148142 5.393729
## Maryland         8.474956  5.853669       8.744170    11.242309 6.972090
## Massachusetts    7.163839  5.282642       6.159327     9.636062 4.045568
## Michigan         6.585197  5.783508       6.011770     9.644444 4.501246
## Minnesota        7.041983  5.744175       6.703591     9.959241 5.488255
## Mississippi      5.618244  7.451404       7.009354     8.182110 5.623936
## Missouri         6.438727  6.328482       5.452462     9.554006 4.521139
## Montana          5.065790  7.773660       7.566820     8.186247 5.622588
## Nebraska         7.149134  7.795013       7.520702    10.678236 5.277986
## Nevada           5.371428  7.580724       7.471178     9.473165 5.213383
## New Hampshire    5.489742  7.227466       7.531054     8.544177 5.376355
## New Jersey       7.156392  5.276604       4.780133    10.036331 4.490014
## New Mexico       0.000000  8.276070       7.836752     9.263648 5.384287
## New York         8.276070  0.000000       7.399846    10.954524 5.304849
## North Carolina   7.836752  7.399846       0.000000    10.768440 5.629634
## North Dakota     9.263648 10.954524      10.768440     0.000000 9.045486
## Ohio             5.384287  5.304849       5.629634     9.045486 0.000000
## Oklahoma         5.784805  6.270010       7.157084     9.703240 3.853859
## Oregon           5.834522  6.660920       6.375622     9.813806 5.183214
## Pennsylvania     6.370792  7.343944       6.379299    10.671674 6.418638
## Rhode Island     5.868621  7.050025       7.127356    10.718173 5.370204
## South Carolina   8.886953  6.493412       6.394603    10.704755 5.683324
## South Dakota     4.954748  8.653921       8.318375     8.687247 6.305190
## Tennessee        7.226178  5.998749       5.926952    10.180302 5.564127
## Texas            5.598018  5.759845       6.498103     9.719807 4.262804
## Utah             8.600500  7.321072       7.536111    11.592308 7.867264
## Vermont          4.922386  8.622720       8.266676     8.419410 6.236120
## Virginia         6.305446  5.688592       6.082835     9.755559 4.150500
## Washington       6.475412  5.590682       5.873186     9.970457 5.567255
## West Virginia    6.091679  7.720694       8.382340     8.910264 5.714675
## Wisconsin        6.707210  6.417604       7.463045     8.903015 5.390976
## Wyoming          5.938386  7.414808       7.071621     8.768103 6.237973
## Grand Total      6.228958  4.495561       5.174696    10.201884 4.585300
##                Oklahoma   Oregon Pennsylvania Rhode Island South Carolina
## Alabama        4.782860 4.064872     7.450679     4.232285       6.555781
## Alaska         6.996699 6.727814     7.121428     7.082969       8.974856
## Alberta        6.270884 6.662351     8.635543     6.718567       9.576543
## Arizona        4.431446 6.005467     5.977805     7.192322       5.262589
## Arkansas       6.146751 3.975158     8.198800     5.380866       7.755976
## Armed Forces   5.576427 5.316895     8.179681     5.610010       8.262035
## California     6.310038 6.624895     6.125095     7.723668       7.205556
## Colorado       6.628926 5.970400     4.381066     6.896976       8.308857
## Connecticut    6.126956 4.312459     6.960579     5.748315       7.034052
## Delaware       6.920917 7.607843     8.056631     7.910543       9.339922
## Florida        5.626614 5.303754     6.424759     5.948382       5.933449
## Georgia        4.192569 5.470112     6.502955     4.828714       7.392903
## Hawaii         5.888685 6.533233     8.631923     8.186827       8.869245
## Idaho          4.916438 6.189566     7.171559     5.970258       8.173157
## Illinois       6.813031 7.438703     7.700079     7.787660       7.520300
## Indiana        5.396766 5.066064     6.071833     5.838866       6.086222
## Iowa           4.960254 5.964436     7.524562     6.901866       7.469876
## Kansas         6.897179 7.154538     8.320207     7.834624       9.817396
## Kentucky       4.694252 5.628527     7.199063     5.756226       8.061831
## Louisiana      5.899778 4.420353     7.156796     5.284769       5.131123
## Maine          5.448801 6.535852     7.458868     6.535732       8.656070
## Maryland       7.121522 8.383445     7.034796     8.722184       8.216247
## Massachusetts  4.742192 5.804385     6.115796     5.941597       6.024039
## Michigan       6.480878 4.784171     6.939652     5.019436       6.590600
## Minnesota      6.672593 4.123454     7.702952     5.414325       5.629351
## Mississippi    6.017652 5.999159     7.743356     5.104582       7.967452
## Missouri       5.189413 4.395396     6.901824     4.639582       5.711664
## Montana        5.354831 5.768759     7.505236     5.712947       8.476217
## Nebraska       6.000114 6.209362     7.583019     6.402327       8.596137
## Nevada         4.569980 5.556000     7.676239     5.756628       8.074329
## New Hampshire  5.306841 4.036584     7.633546     5.879335       7.123741
## New Jersey     6.171824 6.037221     6.400008     6.538849       6.770492
## New Mexico     5.784805 5.834522     6.370792     5.868621       8.886953
## New York       6.270010 6.660920     7.343944     7.050025       6.493412
## North Carolina 7.157084 6.375622     6.379299     7.127356       6.394603
## North Dakota   9.703240 9.813806    10.671674    10.718173      10.704755
## Ohio           3.853859 5.183214     6.418638     5.370204       5.683324
## Oklahoma       0.000000 6.262900     7.063426     6.346805       7.214776
## Oregon         6.262900 0.000000     7.305276     4.171568       6.577092
## Pennsylvania   7.063426 7.305276     0.000000     8.236649       7.615488
## Rhode Island   6.346805 4.171568     8.236649     0.000000       8.658346
## South Carolina 7.214776 6.577092     7.615488     8.658346       0.000000
## South Dakota   5.906544 6.179472     8.723327     6.075385       9.216310
## Tennessee      6.749317 4.260792     7.308986     5.229568       5.944236
## Texas          5.648108 6.008431     5.226836     6.347943       7.397568
## Utah           8.255040 8.276947     8.828539     8.483404       9.513635
## Vermont        6.144165 5.710069     8.773149     5.897447       9.112723
## Virginia       3.650849 5.379701     6.236633     5.567495       7.548536
## Washington     6.613965 5.105833     6.394647     5.228611       7.262601
## West Virginia  6.108545 6.173761     8.711816     6.451735       7.613307
## Wisconsin      5.011600 6.490776     7.253764     6.937719       7.508077
## Wyoming        6.270546 6.298179     7.983332     5.501689       8.420822
## Grand Total    5.864088 5.657728     4.410200     6.750076       6.276222
##                South Dakota Tennessee    Texas      Utah  Vermont Virginia
## Alabama            5.319670  4.606873 6.197279  8.762303 5.149541 4.944948
## Alaska             6.607146  7.524248 7.510870  9.297339 6.761137 6.504468
## Alberta            2.393359  8.198124 8.098935  9.084360 2.498740 6.219916
## Arizona            7.663449  6.013190 5.184887  8.704028 7.621458 3.627328
## Arkansas           3.857254  5.811730 6.629659  8.516727 3.320485 6.087228
## Armed Forces       3.682592  6.814115 7.168956  7.466403 3.941580 5.632139
## California         8.792791  6.406763 5.388334  8.061904 8.708832 5.707211
## Colorado           8.120123  6.874478 4.889869  8.058331 8.042605 5.582686
## Connecticut        5.658362  5.669693 6.006434  6.676023 5.436103 5.402257
## Delaware           6.884977  7.565404 7.884593  8.382922 7.148112 6.793980
## Florida            7.843544  4.748354 4.433368  6.517269 7.686588 4.799523
## Georgia            5.734253  5.705411 5.219959  7.053998 5.819848 3.542946
## Hawaii             7.479454  7.902641 7.829138  9.399234 7.444364 6.734386
## Idaho              5.648095  5.868657 6.090771  7.865078 5.642209 4.879180
## Illinois           8.352783  6.466157 6.668738  8.723810 8.461158 7.167333
## Indiana            5.972264  5.270415 5.543089  5.477808 6.115764 4.559654
## Iowa               6.212878  6.485607 6.001264  6.477314 6.105436 4.543963
## Kansas             6.947364  8.943009 8.209651  8.623943 7.076865 6.050832
## Kentucky           5.475771  5.576714 6.420524  7.300432 5.384558 4.688628
## Louisiana          6.538900  4.917737 5.131024  8.260537 6.440858 5.914366
## Maine              4.413409  7.324797 6.607062  8.981313 4.526146 5.310607
## Maryland           9.439669  8.450909 6.665696 10.223566 9.669427 7.594713
## Massachusetts      7.656587  5.321992 5.386818  8.659978 7.614492 4.246912
## Michigan           6.814901  4.473132 5.324044  8.576313 6.396823 5.554418
## Minnesota          7.328437  3.623236 6.008542  8.438402 6.947430 6.001199
## Mississippi        4.131344  6.327798 7.061935  8.579812 4.289154 6.106232
## Missouri           6.529536  4.268819 5.292006  8.560707 6.470481 5.128780
## Montana            3.676748  6.513385 6.890048  7.840617 3.963550 5.604587
## Nebraska           7.282063  6.486458 6.905791  9.454008 7.125633 4.853765
## Nevada             4.889343  7.679118 6.450885  8.107921 5.330111 5.857743
## New Hampshire      4.197512  5.918545 6.633053  8.117123 3.906106 5.438581
## New Jersey         7.322902  5.684491 5.511662  7.277510 7.174625 4.429281
## New Mexico         4.954748  7.226178 5.598018  8.600500 4.922386 6.305446
## New York           8.653921  5.998749 5.759845  7.321072 8.622720 5.688592
## North Carolina     8.318375  5.926952 6.498103  7.536111 8.266676 6.082835
## North Dakota       8.687247 10.180302 9.719807 11.592308 8.419410 9.755559
## Ohio               6.305190  5.564127 4.262804  7.867264 6.236120 4.150500
## Oklahoma           5.906544  6.749317 5.648108  8.255040 6.144165 3.650849
## Oregon             6.179472  4.260792 6.008431  8.276947 5.710069 5.379701
## Pennsylvania       8.723327  7.308986 5.226836  8.828539 8.773149 6.236633
## Rhode Island       6.075385  5.229568 6.347943  8.483404 5.897447 5.567495
## South Carolina     9.216310  5.944236 7.397568  9.513635 9.112723 7.548536
## South Dakota       0.000000  7.806621 7.928437  8.557163 1.265816 6.615854
## Tennessee          7.806621  0.000000 5.845713  8.210351 7.381379 5.820073
## Texas              7.928437  5.845713 0.000000  7.716839 7.691491 5.240816
## Utah               8.557163  8.210351 7.716839  0.000000 8.793144 7.945931
## Vermont            1.265816  7.381379 7.691491  8.793144 0.000000 6.566671
## Virginia           6.615854  5.820073 5.240816  7.945931 6.566671 0.000000
## Washington         6.949806  5.222686 6.022337  6.209611 7.058193 5.543376
## West Virginia      3.798077  7.422068 7.870934  9.315837 3.812299 6.841551
## Wisconsin          6.351325  6.953639 5.190576  7.841146 6.540098 6.051127
## Wyoming            5.196751  6.652550 6.854230  6.708951 5.648347 6.521503
## Grand Total        7.977880  5.343531 4.025021  7.504707 7.865734 5.041671
##                Washington West Virginia Wisconsin  Wyoming Grand Total
## Alabama          5.716118      4.407156  6.120516 4.915825    6.218044
## Alaska           7.197493      7.237112  6.813791 6.411280    6.649416
## Alberta          7.648839      4.366555  7.019624 6.064700    7.984547
## Arizona          6.318643      6.781851  5.826823 7.182683    4.274130
## Arkansas         5.974328      4.848998  5.454567 5.037459    6.983158
## Armed Forces     5.627650      4.866620  5.995199 4.899935    7.271430
## California       6.853838      8.845652  6.370484 7.770176    2.442488
## Colorado         6.057213      8.273067  7.417557 7.430232    4.109117
## Connecticut      5.590082      5.056785  6.862335 5.426891    5.414111
## Delaware         6.302715      7.814470  7.082930 6.325067    7.635680
## Florida          4.921293      7.831506  5.833695 6.679687    3.667190
## Georgia          5.142458      5.634109  6.279280 5.185658    5.089513
## Hawaii           8.710108      7.711825  7.694510 7.762876    7.333753
## Idaho            6.069777      6.437182  5.308922 6.195199    6.373841
## Illinois         7.090325      8.037606  6.725349 6.934846    5.266276
## Indiana          3.830936      6.165578  5.284446 4.578402    4.921929
## Iowa             6.664780      6.860840  6.245025 6.896238    5.891074
## Kansas           6.329396      7.711189  7.953246 8.005924    7.772135
## Kentucky         5.919382      6.070705  6.097142 6.064219    6.443837
## Louisiana        5.562652      5.818086  5.311139 5.393941    5.507967
## Maine            6.613086      5.441244  5.189702 5.563723    6.691132
## Maryland         7.850963      9.255390  5.931201 8.299394    5.075635
## Massachusetts    5.675188      7.029111  5.924734 6.739440    4.526745
## Michigan         5.106184      6.093132  6.477349 6.319361    5.275256
## Minnesota        5.338240      7.228503  5.868447 6.656422    5.802195
## Mississippi      6.103148      4.219397  6.167304 3.849856    7.003535
## Missouri         5.563479      6.275096  4.845933 5.044458    5.443779
## Montana          5.735890      5.100971  5.675933 4.719607    7.067734
## Nebraska         7.012228      7.654238  7.611519 8.279431    6.657441
## Nevada           6.296989      5.875271  4.729871 4.955522    6.891992
## New Hampshire    6.319814      4.562432  5.810459 5.926614    6.812184
## New Jersey       5.592976      7.293378  6.648407 6.706337    4.143404
## New Mexico       6.475412      6.091679  6.707210 5.938386    6.228958
## New York         5.590682      7.720694  6.417604 7.414808    4.495561
## North Carolina   5.873186      8.382340  7.463045 7.071621    5.174696
## North Dakota     9.970457      8.910264  8.903015 8.768103   10.201884
## Ohio             5.567255      5.714675  5.390976 6.237973    4.585300
## Oklahoma         6.613965      6.108545  5.011600 6.270546    5.864088
## Oregon           5.105833      6.173761  6.490776 6.298179    5.657728
## Pennsylvania     6.394647      8.711816  7.253764 7.983332    4.410200
## Rhode Island     5.228611      6.451735  6.937719 5.501689    6.750076
## South Carolina   7.262601      7.613307  7.508077 8.420822    6.276222
## South Dakota     6.949806      3.798077  6.351325 5.196751    7.977880
## Tennessee        5.222686      7.422068  6.953639 6.652550    5.343531
## Texas            6.022337      7.870934  5.190576 6.854230    4.025021
## Utah             6.209611      9.315837  7.841146 6.708951    7.504707
## Vermont          7.058193      3.812299  6.540098 5.648347    7.865734
## Virginia         5.543376      6.841551  6.051127 6.521503    5.041671
## Washington       0.000000      7.218800  6.229651 5.286578    5.456135
## West Virginia    7.218800      0.000000  7.302107 5.788025    7.801763
## Wisconsin        6.229651      7.302107  0.000000 5.756116    5.973850
## Wyoming          5.286578      5.788025  5.756116 0.000000    6.981136
## Grand Total      5.456135      7.801763  5.973850 6.981136    0.000000
clusterdata<-hclust(dm,method="average")
plot(clusterdata, hang=-1, cex=0.7, main="Average Linkage Cluster")

#the cluster does not give too much insight at the first look.

dm1<-dist(datanew,method="manhattan")
## Warning in dist(datanew, method = "manhattan"): NAs introduced by coercion
#using the euclidean method of calculating distance.
as.matrix(dm1)
##                 Alabama   Alaska   Alberta  Arizona  Arkansas Armed Forces
## Alabama         0.00000 20.31159 18.976128 20.03722 11.253086    16.669843
## Alaska         20.31159  0.00000 17.328556 24.46068 20.615944    19.769633
## Alberta        18.97613 17.32856  0.000000 29.29593 12.318433     9.851319
## Arizona        20.03722 24.46068 29.295931  0.00000 25.425958    24.853712
## Arkansas       11.25309 20.61594 12.318433 25.42596  0.000000     9.305616
## Armed Forces   16.66984 19.76963  9.851319 24.85371  9.305616     0.000000
## California     27.83711 27.85786 37.419206 16.49585 30.984383    33.544855
## Colorado       23.98191 22.10908 32.711078 21.53585 27.440321    28.944315
## Connecticut    16.59170 20.85104 21.684465 20.45611 19.107739    17.997968
## Delaware       21.41439 18.63402 18.290272 28.98330 20.883415    17.326531
## Florida        21.38199 29.47949 34.584032 19.08875 25.886872    27.480478
## Georgia        14.89713 24.34487 19.546543 18.49043 20.213409    17.798748
## Hawaii         20.51719 21.50021 21.158729 28.31150 20.665464    20.684987
## Idaho          16.62003 13.80018 18.526529 21.55902 16.943322    14.955615
## Illinois       18.12264 27.61557 33.527983 22.61051 24.888633    26.606480
## Indiana        19.19856 20.02672 23.152353 16.00733 19.631145    19.025244
## Iowa           21.07542 25.33797 20.169170 22.45496 20.883869    16.025238
## Kansas         19.04476 25.42955 18.926510 25.67269 19.805012    16.278450
## Kentucky       13.68269 18.17608 18.049273 23.47098 15.675881    15.396904
## Louisiana      13.72988 21.21427 25.403679 17.37079 17.110150    20.115501
## Maine          16.59319 14.06942 11.369237 22.18379 15.536066    17.236455
## Maryland       27.28224 29.61818 37.595864 22.02927 33.385943    31.641413
## Massachusetts  16.94373 22.27054 29.937018 12.98014 23.747205    23.945520
## Michigan       16.42389 22.97566 24.959743 18.19643 19.105590    24.695096
## Minnesota      15.20893 26.63239 28.038975 19.35506 16.909497    18.991979
## Mississippi    10.86045 13.64060 12.723235 25.38104 11.821992    14.225640
## Missouri       11.24909 23.17651 25.297653 18.38342 16.582356    20.797560
## Montana        14.57501 10.29458 10.726289 25.03126 12.055906     9.762548
## Nebraska       20.42491 24.40204 19.105831 22.87440 19.777661    19.384824
## Nevada         16.86483 23.25291 14.894116 24.87526 14.861262    13.497930
## New Hampshire  11.19578 19.60041 12.830140 23.67208  8.775531    11.356880
## New Jersey     24.50448 28.79231 28.160784 15.49387 24.696489    24.433414
## New Mexico     17.22819 21.86912 17.350830 27.60650 15.746460    17.239201
## New York       22.17275 31.87838 35.904317 18.90562 27.464108    29.930044
## North Carolina 23.82039 27.87010 32.145508 20.48682 25.990708    28.039401
## North Dakota   26.79011 20.43800 15.872132 34.48396 16.864214    16.934285
## Ohio           16.19335 24.08539 25.341332 13.09612 20.804178    21.606122
## Oklahoma       15.64526 22.95873 20.208620 17.10533 20.148455    16.787325
## Oregon         14.71912 23.65355 23.250168 20.95224 13.683801    17.614876
## Pennsylvania   27.42519 24.27405 34.086967 18.62427 31.462818    30.945841
## Rhode Island   11.98033 22.59860 18.757365 26.62247 15.724123    13.501457
## South Carolina 21.36749 29.71385 35.874570 16.51613 26.213861    27.686003
## South Dakota   15.89104 17.64829  3.085087 30.87204  9.233346     6.766233
## Tennessee      14.93753 26.36773 31.085393 23.43490 19.555891    22.618587
## Texas          22.00008 27.32075 33.432055 17.12330 25.463996    26.886259
## Utah           28.36697 30.13563 27.382559 29.15756 26.364776    19.700857
## Vermont        14.90946 19.45909  4.066663 30.32046  8.251770     9.256787
## Virginia       16.15828 22.15049 22.245499 12.01769 20.438974    18.161788
## Washington     19.72309 26.50825 28.406444 23.44163 19.667529    19.228926
## West Virginia  13.35671 20.93987  8.681979 26.67069 12.935840    11.724147
## Wisconsin      19.00884 24.16372 23.290581 22.49466 16.917229    18.367402
## Wyoming        14.82612 17.59521 16.294859 28.29545 12.111368    12.173483
## Grand Total    24.97941 27.88030 36.722376 15.50024 29.868455    31.835936
##                California Colorado Connecticut Delaware  Florida  Georgia
## Alabama         27.837112 23.98191    16.59170 21.41439 21.38199 14.89713
## Alaska          27.857857 22.10908    20.85104 18.63402 29.47949 24.34487
## Alberta         37.419206 32.71108    21.68446 18.29027 34.58403 19.54654
## Arizona         16.495854 21.53585    20.45611 28.98330 19.08875 18.49043
## Arkansas        30.984383 27.44032    19.10774 20.88342 25.88687 20.21341
## Armed Forces    33.544855 28.94432    17.99797 17.32653 27.48048 17.79875
## California       0.000000 17.80415    24.05169 32.19899 14.48961 19.71631
## Colorado        17.804149  0.00000    17.85917 31.52071 19.34793 18.24462
## Connecticut     24.051692 17.85917     0.00000 23.20782 20.45297 18.80308
## Delaware        32.198987 31.52071    23.20782  0.00000 25.35557 25.92145
## Florida         14.489607 19.34793    20.45297 25.35557  0.00000 20.18585
## Georgia         19.716307 18.24462    18.80308 25.92145 20.18585  0.00000
## Hawaii          31.249971 25.85312    20.76613 28.03228 33.91679 22.97599
## Idaho           27.313412 22.53185    21.40351 19.66853 23.84919 18.47902
## Illinois        19.656711 23.49202    22.74554 28.58156 21.38261 21.09052
## Indiana         18.352940 17.62953    15.38526 20.70050 13.72204 16.07314
## Iowa            25.346863 24.66780    17.94506 21.11135 17.99378 19.21283
## Kansas          33.614074 24.62123    21.86597 21.96788 26.12555 21.16920
## Kentucky        27.427301 21.58531    19.11506 19.34188 22.93777 16.99053
## Louisiana       25.174249 24.99546    16.89411 23.62508 17.69450 20.93653
## Maine           31.085668 26.73773    23.37173 18.72852 28.34321 18.85956
## Maryland        19.436443 26.30559    28.90944 32.09412 27.65922 26.66943
## Massachusetts   17.890191 22.02160    22.41205 28.84306 18.83701 16.67303
## Michigan        23.583510 24.73167    19.92087 25.10644 15.57280 19.55239
## Minnesota       24.021632 23.39793    22.05746 29.20504 16.96736 16.51942
## Mississippi     33.077882 26.89285    18.73656 16.85547 25.66360 19.32683
## Missouri        25.373026 25.72865    21.31172 24.71223 18.76106 18.55704
## Montana         32.108878 26.49027    18.13859  8.97311 25.93551 20.42840
## Nebraska        29.885913 25.34348    26.89710 28.96046 25.18657 16.28852
## Nevada          31.028628 30.03513    21.99694 17.21436 26.80011 19.39795
## New Hampshire   29.466107 25.01625    13.93998 17.84909 23.52161 19.62448
## New Jersey      17.338210 21.51039    19.87881 26.80485 10.96405 17.16194
## New Mexico      31.256783 18.74357    20.78455 21.10924 27.51726 17.06673
## New York        16.486956 22.95580    22.16525 28.61039 14.74332 21.09924
## North Carolina  24.211170 21.35864    20.09060 30.34934 15.49093 23.70587
## North Dakota    42.690181 39.52857    28.11173 23.90885 38.57023 31.97686
## Ohio            18.622580 20.22628    21.78695 25.50626 14.70132 12.94288
## Oklahoma        23.646301 22.76188    20.55220 19.40093 20.33589 14.10231
## Oregon          25.844064 22.42428    15.36555 23.53970 21.36216 19.87207
## Pennsylvania    19.776062 14.28653    23.65021 30.70528 19.34592 23.61336
## Rhode Island    30.898150 25.19563    19.74667 20.86518 22.28590 17.27597
## South Carolina  26.786422 31.04362    25.74181 33.05805 22.33636 28.10998
## South Dakota    38.940838 32.56843    20.61908 15.20519 32.84385 21.12265
## Tennessee       24.522614 23.96986    22.19956 27.67095 14.95354 20.32039
## Texas           16.967069 14.88262    21.63475 30.20784 16.06418 19.56021
## Utah            30.731340 29.26040    20.35318 23.16074 22.94255 24.05124
## Vermont         37.959261 31.58686    18.12852 17.01598 31.86227 21.34156
## Virginia        20.817065 17.70627    18.30343 21.79675 17.96680 12.89627
## Washington      24.107047 20.01224    18.22826 23.04260 16.26723 16.54781
## West Virginia   37.968529 32.20319    17.20112 19.18806 31.94516 19.52659
## Wisconsin       24.715309 26.57896    22.80601 22.52253 21.64806 20.96400
## Wyoming         31.297319 26.53762    17.64555 18.61169 24.87123 18.44276
## Grand Total      9.547551 13.78615    20.24311 31.04183 12.20837 19.11311
##                  Hawaii     Idaho Illinois  Indiana     Iowa   Kansas
## Alabama        20.51719 16.620029 18.12264 19.19856 21.07542 19.04476
## Alaska         21.50021 13.800182 27.61557 20.02672 25.33797 25.42955
## Alberta        21.15873 18.526529 33.52798 23.15235 20.16917 18.92651
## Arizona        28.31150 21.559023 22.61051 16.00733 22.45496 25.67269
## Arkansas       20.66546 16.943322 24.88863 19.63114 20.88387 19.80501
## Armed Forces   20.68499 14.955615 26.60648 19.02524 16.02524 16.27845
## California     31.24997 27.313412 19.65671 18.35294 25.34686 33.61407
## Colorado       25.85312 22.531852 23.49202 17.62953 24.66780 24.62123
## Connecticut    20.76613 21.403506 22.74554 15.38526 17.94506 21.86597
## Delaware       28.03228 19.668532 28.58156 20.70050 21.11135 21.96788
## Florida        33.91679 23.849191 21.38261 13.72204 17.99378 26.12555
## Georgia        22.97599 18.479015 21.09052 16.07314 19.21283 21.16920
## Hawaii          0.00000 21.255773 30.11163 26.81830 23.22623 21.80665
## Idaho          21.25577  0.000000 21.24579 17.78627 16.54876 18.97211
## Illinois       30.11163 21.245787  0.00000 19.28164 23.57934 28.41961
## Indiana        26.81830 17.786275 19.28164  0.00000 16.39992 20.13191
## Iowa           23.22623 16.548765 23.57934 16.39992  0.00000 19.67928
## Kansas         21.80665 18.972112 28.41961 20.13191 19.67928  0.00000
## Kentucky       19.01004  5.813247 22.29605 15.57944 15.62478 18.16852
## Louisiana      28.83174 21.665863 20.12429 16.45617 21.88322 26.42722
## Maine          21.81556 13.379910 27.21294 19.27823 21.43321 16.30213
## Maryland       34.42399 27.686414 18.29962 26.65376 29.59102 33.86820
## Massachusetts  28.96748 20.636658 25.17128 19.19981 23.93183 29.72997
## Michigan       34.23983 23.267377 25.76463 15.53368 25.70222 29.19954
## Minnesota      30.16487 18.665751 20.91146 17.72711 22.99515 25.48711
## Mississippi    24.51224 13.360119 23.43077 17.51969 22.66632 20.57306
## Missouri       24.08812 18.471077 20.20916 19.07509 22.35687 23.59208
## Montana        20.32269 10.973600 24.66825 15.89244 17.00375 16.64658
## Nebraska       25.70911 18.338105 32.86218 22.94511 25.07976 22.55969
## Nevada         19.14417 22.682092 24.93341 20.52238 20.81337 15.96309
## New Hampshire  16.58246 15.868859 21.52054 15.91823 14.96273 15.86129
## New Jersey     35.87518 25.826009 25.86952 14.49820 19.83755 27.63810
## New Mexico     22.01889 20.676499 26.08672 24.05194 21.47296 18.70043
## New York       34.92512 24.967661 17.89718 18.56525 22.11946 28.95792
## North Carolina 31.80336 26.086674 29.57829 18.28417 22.82181 24.44242
## North Dakota   28.06300 19.709513 34.15768 27.47922 27.04154 27.90761
## Ohio           27.35013 18.688207 22.46850 14.95993 19.14530 20.63981
## Oklahoma       18.07430 17.390940 22.02276 17.80649 16.84445 16.10874
## Oregon         21.11336 21.550420 26.51046 17.93150 23.29999 21.06175
## Pennsylvania   33.26615 25.087889 26.78512 17.79488 27.57057 28.75057
## Rhode Island   26.03658 18.386104 26.33174 19.94497 22.73166 19.85495
## South Carolina 31.55208 25.901139 24.07543 21.71202 25.05188 30.69713
## South Dakota   19.58262 15.446663 30.44290 22.79963 20.20014 15.84142
## Tennessee      28.08262 19.216529 21.23572 18.36287 22.50048 28.25386
## Texas          29.79658 20.637834 20.90209 17.72862 22.18526 26.93913
## Utah           31.32646 24.562237 29.11573 15.06359 16.42519 23.31042
## Vermont        19.88444 16.051755 31.05321 23.40995 19.30147 17.65222
## Virginia       23.95367 15.885167 24.62242 17.58760 17.33971 17.85940
## Washington     31.90381 20.869136 23.04593 11.91217 22.59049 23.21125
## West Virginia  21.77614 18.733029 27.85058 21.87169 21.16662 19.54392
## Wisconsin      23.98103 16.837189 19.10185 18.66048 18.30438 21.06126
## Wyoming        22.06156 17.616504 21.21878 16.43106 23.07958 22.75935
## Grand Total    32.22043 26.009177 19.07237 17.52780 25.22992 30.23127
##                 Kentucky Louisiana    Maine Maryland Massachusetts
## Alabama        13.682690 13.729880 16.59319 27.28224      16.94373
## Alaska         18.176085 21.214267 14.06942 29.61818      22.27054
## Alberta        18.049273 25.403679 11.36924 37.59586      29.93702
## Arizona        23.470977 17.370789 22.18379 22.02927      12.98014
## Arkansas       15.675881 17.110150 15.53607 33.38594      23.74721
## Armed Forces   15.396904 20.115501 17.23645 31.64141      23.94552
## California     27.427301 25.174249 31.08567 19.43644      17.89019
## Colorado       21.585310 24.995458 26.73773 26.30559      22.02160
## Connecticut    19.115065 16.894112 23.37173 28.90944      22.41205
## Delaware       19.341883 23.625079 18.72852 32.09412      28.84306
## Florida        22.937768 17.694500 28.34321 27.65922      18.83701
## Georgia        16.990526 20.936534 18.85956 26.66943      16.67303
## Hawaii         19.010037 28.831739 21.81556 34.42399      28.96748
## Idaho           5.813247 21.665863 13.37991 27.68641      20.63666
## Illinois       22.296045 20.124292 27.21294 18.29962      25.17128
## Indiana        15.579444 16.456169 19.27823 26.65376      19.19981
## Iowa           15.624784 21.883219 21.43321 29.59102      23.93183
## Kansas         18.168521 26.427221 16.30213 33.86820      29.72997
## Kentucky        0.000000 24.167575 16.13191 30.39646      22.49108
## Louisiana      24.167575  0.000000 22.66669 27.34237      18.19219
## Maine          16.131907 22.666694  0.00000 28.97027      24.01426
## Maryland       30.396461 27.342371 28.97027  0.00000      22.28551
## Massachusetts  22.491077 18.192187 24.01426 22.28551       0.00000
## Michigan       22.069566 15.695216 22.13798 30.19639      16.34396
## Minnesota      20.655857 13.562278 23.70820 27.46184      18.06321
## Mississippi    14.216775 15.807413 13.39877 31.23475      22.12885
## Missouri       19.634419  9.886958 19.76804 25.58292      17.57096
## Montana        11.910478 17.559720 12.11758 32.00454      24.29394
## Nebraska       18.020195 24.918190 19.89253 35.90900      22.00145
## Nevada         21.878500 17.350608 17.51629 26.64722      25.89393
## New Hampshire  13.158015 16.290078 17.75728 29.68881      25.30187
## New Jersey     25.000897 18.938337 26.00960 29.55171      19.89498
## New Mexico     19.872907 20.936821 17.48397 30.95257      26.97997
## New York       24.970461 24.183245 28.50320 22.36703      19.74489
## North Carolina 26.266971 20.101937 26.60087 32.77478      21.02880
## North Dakota   23.264790 27.163363 16.95333 39.55659      34.41559
## Ohio           19.877904 13.345140 19.84673 23.89650      13.37501
## Oklahoma       16.050790 20.735301 18.51293 23.79762      17.64300
## Oregon         18.302189 16.079873 22.88313 28.64533      22.12396
## Pennsylvania   24.784569 26.217482 27.31149 25.38533      19.18344
## Rhode Island   17.949517 17.650260 21.57964 30.97822      21.01274
## South Carolina 26.108441 16.615363 29.14053 30.06218      21.47217
## South Dakota   14.964186 22.318592 11.83538 36.50430      29.94867
## Tennessee      17.456352 18.542241 27.14500 32.31315      19.94648
## Texas          21.609298 18.694663 24.59627 21.74740      19.87798
## Utah           23.655848 24.392268 27.43905 37.02819      29.97592
## Vermont        15.574500 21.337016 13.64617 37.11461      28.96709
## Virginia       16.092930 21.361001 18.82310 26.00000      15.75056
## Washington     19.999906 18.608655 25.60910 28.23606      20.61270
## West Virginia  17.469507 20.011711 15.29046 33.81117      27.14287
## Wisconsin      17.965180 19.106925 17.75196 19.44736      22.57782
## Wyoming        17.831940 18.569221 18.06286 31.05164      24.10689
## Grand Total    26.376060 21.326594 28.68414 18.95118      15.97344
##                Michigan Minnesota Mississippi  Missouri   Montana Nebraska
## Alabama        16.42389  15.20893   10.860455 11.249085 14.575009 20.42491
## Alaska         22.97566  26.63239   13.640599 23.176513 10.294580 24.40204
## Alberta        24.95974  28.03898   12.723235 25.297653 10.726289 19.10583
## Arizona        18.19643  19.35506   25.381037 18.383416 25.031256 22.87440
## Arkansas       19.10559  16.90950   11.821992 16.582356 12.055906 19.77766
## Armed Forces   24.69510  18.99198   14.225640 20.797560  9.762548 19.38482
## California     23.58351  24.02163   33.077882 25.373026 32.108878 29.88591
## Colorado       24.73167  23.39793   26.892855 25.728652 26.490270 25.34348
## Connecticut    19.92087  22.05746   18.736565 21.311721 18.138591 26.89710
## Delaware       25.10644  29.20504   16.855471 24.712228  8.973110 28.96046
## Florida        15.57280  16.96736   25.663597 18.761056 25.935511 25.18657
## Georgia        19.55239  16.51942   19.326835 18.557037 20.428402 16.28852
## Hawaii         34.23983  30.16487   24.512238 24.088120 20.322691 25.70911
## Idaho          23.26738  18.66575   13.360119 18.471077 10.973600 18.33810
## Illinois       25.76463  20.91146   23.430774 20.209162 24.668246 32.86218
## Indiana        15.53368  17.72711   17.519693 19.075089 15.892443 22.94511
## Iowa           25.70222  22.99515   22.666320 22.356874 17.003749 25.07976
## Kansas         29.19954  25.48711   20.573064 23.592079 16.646583 22.55969
## Kentucky       22.06957  20.65586   14.216775 19.634419 11.910478 18.02020
## Louisiana      15.69522  13.56228   15.807413  9.886958 17.559720 24.91819
## Maine          22.13798  23.70820   13.398766 19.768037 12.117584 19.89253
## Maryland       30.19639  27.46184   31.234748 25.582918 32.004543 35.90900
## Massachusetts  16.34396  18.06321   22.128845 17.570957 24.293935 22.00145
## Michigan        0.00000  18.97307   18.330055 17.295669 21.250802 22.06846
## Minnesota      18.97307   0.00000   19.006677 11.670552 22.020667 21.81826
## Mississippi    18.33006  19.00668    0.000000 14.367863  8.772148 23.57318
## Missouri       17.29567  11.67055   14.367863  0.000000 19.575707 22.46299
## Montana        21.25080  22.02067    8.772148 19.575707  0.000000 19.98735
## Nebraska       22.06846  21.81826   23.573181 22.462987 19.987350  0.00000
## Nevada         24.48795  23.23577   16.496898 17.056969 14.332849 24.85721
## New Hampshire  20.67505  18.13615   13.070988 16.715141 10.249421 20.03363
## New Jersey     15.59395  18.93374   24.129539 22.048016 25.902479 20.62532
## New Mexico     25.59350  20.41367   16.428519 20.433583 14.609112 22.84171
## New York       21.03085  21.11314   29.107532 23.121281 29.388208 31.44769
## North Carolina 20.07704  24.48891   23.471044 19.177786 26.981741 27.53513
## North Dakota   32.23846  30.56713   17.926990 29.167290 15.081344 31.33164
## Ohio           15.82221  15.60925   20.362188 15.478290 20.309250 17.22541
## Oklahoma       22.11118  20.66654   18.410159 17.488058 15.585389 19.55209
## Oregon         17.83384  14.50397   19.307735 14.576933 17.763087 20.30191
## Pennsylvania   21.58466  25.84174   28.272757 24.009629 28.014580 27.33394
## Rhode Island   18.68117  17.56240   14.578858 16.325355 15.664008 20.05465
## South Carolina 23.48297  18.23607   26.448768 16.984732 27.277547 31.23489
## South Dakota   25.22840  24.95389    9.638148 22.212566  7.641202 20.68194
## Tennessee      17.04879  12.99946   20.943554 16.410825 20.887768 18.98002
## Texas          19.90416  20.29458   26.385003 20.489550 25.783776 25.74539
## Utah           28.72772  26.70017   25.604083 27.851707 20.196540 31.74879
## Vermont        24.24683  23.97231   10.248461 21.230990  9.452000 20.90085
## Virginia       20.21054  19.16671   21.244153 17.880015 19.179754 18.44795
## Washington     17.97445  17.11091   20.889671 20.958774 18.794477 23.40532
## West Virginia  21.45402  23.97719   11.356288 22.411339 11.902261 23.24661
## Wisconsin      21.79468  19.33824   19.895718 14.353877 18.377492 25.70756
## Wyoming        21.03603  21.19326   10.406967 18.927989 12.146991 26.38035
## Grand Total    19.21670  20.56181   30.550270 21.825512 31.204740 27.57631
##                  Nevada New Hampshire New Jersey New Mexico New York
## Alabama        16.86483     11.195783   24.50448   17.22819 22.17275
## Alaska         23.25291     19.600407   28.79231   21.86912 31.87838
## Alberta        14.89412     12.830140   28.16078   17.35083 35.90432
## Arizona        24.87526     23.672080   15.49387   27.60650 18.90562
## Arkansas       14.86126      8.775531   24.69649   15.74646 27.46411
## Armed Forces   13.49793     11.356880   24.43341   17.23920 29.93004
## California     31.02863     29.466107   17.33821   31.25678 16.48696
## Colorado       30.03513     25.016249   21.51039   18.74357 22.95580
## Connecticut    21.99694     13.939978   19.87881   20.78455 22.16525
## Delaware       17.21436     17.849090   26.80485   21.10924 28.61039
## Florida        26.80011     23.521611   10.96405   27.51726 14.74332
## Georgia        19.39795     19.624475   17.16194   17.06673 21.09924
## Hawaii         19.14417     16.582460   35.87518   22.01889 34.92512
## Idaho          22.68209     15.868859   25.82601   20.67650 24.96766
## Illinois       24.93341     21.520535   25.86952   26.08672 17.89718
## Indiana        20.52238     15.918227   14.49820   24.05194 18.56525
## Iowa           20.81337     14.962728   19.83755   21.47296 22.11946
## Kansas         15.96309     15.861287   27.63810   18.70043 28.95792
## Kentucky       21.87850     13.158015   25.00090   19.87291 24.97046
## Louisiana      17.35061     16.290078   18.93834   20.93682 24.18324
## Maine          17.51629     17.757285   26.00960   17.48397 28.50320
## Maryland       26.64722     29.688806   29.55171   30.95257 22.36703
## Massachusetts  25.89393     25.301872   19.89498   26.97997 19.74489
## Michigan       24.48795     20.675048   15.59395   25.59350 21.03085
## Minnesota      23.23577     18.136148   18.93374   20.41367 21.11314
## Mississippi    16.49690     13.070988   24.12954   16.42852 29.10753
## Missouri       17.05697     16.715141   22.04802   20.43358 23.12128
## Montana        14.33285     10.249421   25.90248   14.60911 29.38821
## Nebraska       24.85721     20.033634   20.62532   22.84171 31.44769
## Nevada          0.00000     12.602787   26.65911   15.39728 27.84806
## New Hampshire  12.60279      0.000000   25.09073   16.49895 25.54721
## New Jersey     26.65911     25.090729    0.00000   26.95692 18.74886
## New Mexico     15.39728     16.498955   26.95692    0.00000 31.83203
## New York       27.84806     25.547209   18.74886   31.83203  0.00000
## North Carolina 26.74396     26.485459   17.28451   27.37849 23.78512
## North Dakota   24.35782     19.411240   36.17295   25.03452 40.97641
## Ohio           19.60470     20.267176   17.28569   18.73289 18.65510
## Oklahoma       12.44683     14.008411   22.71931   17.29945 22.38419
## Oregon         18.83432     12.255883   23.40179   18.15087 26.16148
## Pennsylvania   29.51528     27.301017   19.58123   25.92925 25.74294
## Rhode Island   16.50944     16.586046   25.33159   19.20174 27.30530
## South Carolina 26.77426     24.716460   24.79204   30.63842 23.36747
## South Dakota   11.80903     11.254031   29.73689   14.26574 35.29226
## Tennessee      29.15476     20.512028   19.08787   27.10566 22.90558
## Texas          24.84743     24.169618   19.07810   22.09443 19.24495
## Utah           22.21335     22.398358   24.68063   26.28197 26.06734
## Vermont        13.61983     10.355367   28.75532   14.48465 34.31069
## Virginia       20.66282     17.386031   15.83488   22.40265 18.08918
## Washington     23.19871     20.146302   17.44250   24.86067 21.65884
## West Virginia  15.51152     11.787126   28.78391   17.75708 29.69537
## Wisconsin      13.16067     15.004906   24.67580   20.91936 21.96888
## Wyoming        14.48957     16.128630   24.85597   19.24998 25.45214
## Grand Total    30.17881     28.208958   14.35124   26.68917 14.90540
##                North Carolina North Dakota     Ohio Oklahoma   Oregon
## Alabama              23.82039     26.79011 16.19335 15.64526 14.71912
## Alaska               27.87010     20.43800 24.08539 22.95873 23.65355
## Alberta              32.14551     15.87213 25.34133 20.20862 23.25017
## Arizona              20.48682     34.48396 13.09612 17.10533 20.95224
## Arkansas             25.99071     16.86421 20.80418 20.14846 13.68380
## Armed Forces         28.03940     16.93429 21.60612 16.78732 17.61488
## California           24.21117     42.69018 18.62258 23.64630 25.84406
## Colorado             21.35864     39.52857 20.22628 22.76188 22.42428
## Connecticut          20.09060     28.11173 21.78695 20.55220 15.36555
## Delaware             30.34934     23.90885 25.50626 19.40093 23.53970
## Florida              15.49093     38.57023 14.70132 20.33589 21.36216
## Georgia              23.70587     31.97686 12.94288 14.10231 19.87207
## Hawaii               31.80336     28.06300 27.35013 18.07430 21.11336
## Idaho                26.08667     19.70951 18.68821 17.39094 21.55042
## Illinois             29.57829     34.15768 22.46850 22.02276 26.51046
## Indiana              18.28417     27.47922 14.95993 17.80649 17.93150
## Iowa                 22.82181     27.04154 19.14530 16.84445 23.29999
## Kansas               24.44242     27.90761 20.63981 16.10874 21.06175
## Kentucky             26.26697     23.26479 19.87790 16.05079 18.30219
## Louisiana            20.10194     27.16336 13.34514 20.73530 16.07987
## Maine                26.60087     16.95333 19.84673 18.51293 22.88313
## Maryland             32.77478     39.55659 23.89650 23.79762 28.64533
## Massachusetts        21.02880     34.41559 13.37501 17.64300 22.12396
## Michigan             20.07704     32.23846 15.82221 22.11118 17.83384
## Minnesota            24.48891     30.56713 15.60925 20.66654 14.50397
## Mississippi          23.47104     17.92699 20.36219 18.41016 19.30773
## Missouri             19.17779     29.16729 15.47829 17.48806 14.57693
## Montana              26.98174     15.08134 20.30925 15.58539 17.76309
## Nebraska             27.53513     31.33164 17.22541 19.55209 20.30191
## Nevada               26.74396     24.35782 19.60470 12.44683 18.83432
## New Hampshire        26.48546     19.41124 20.26718 14.00841 12.25588
## New Jersey           17.28451     36.17295 17.28569 22.71931 23.40179
## New Mexico           27.37849     25.03452 18.73289 17.29945 18.15087
## New York             23.78512     40.97641 18.65510 22.38419 26.16148
## North Carolina        0.00000     38.31772 18.60779 24.11510 23.72566
## North Dakota         38.31772      0.00000 31.06861 28.99744 30.54802
## Ohio                 18.60779     31.06861  0.00000 12.67299 19.05748
## Oklahoma             24.11510     28.99744 12.67299  0.00000 19.66210
## Oregon               23.72566     30.54802 19.05748 19.66210  0.00000
## Pennsylvania         20.77417     39.29046 22.11209 23.15760 26.31792
## Rhode Island         24.83108     29.96830 19.91441 19.60264 14.62957
## South Carolina       20.28870     35.69142 18.27370 24.50275 21.90069
## South Dakota         31.35186     14.29602 24.41830 18.53653 20.16508
## Tennessee            22.12095     33.69589 20.15045 24.00527 15.66301
## Texas                21.47042     34.95737 14.92164 18.64663 22.40386
## Utah                 25.94637     33.20933 25.85799 25.15568 25.11153
## Vermont              30.37029     14.59784 24.63721 20.34733 19.18350
## Virginia             19.20740     32.38027 15.35873 12.77366 18.97065
## Washington           20.41562     31.69566 20.08049 24.21716 18.30103
## West Virginia        32.49550     18.63750 21.15786 18.45666 20.34870
## Wisconsin            24.12493     23.59596 20.41626 14.89053 22.30860
## Wyoming              23.73809     20.31911 23.30401 20.32748 22.65924
## Grand Total          19.06656     41.80287 16.63770 23.27713 23.62923
##                Pennsylvania Rhode Island South Carolina South Dakota
## Alabama            27.42519     11.98033       21.36749    15.891041
## Alaska             24.27405     22.59860       29.71385    17.648287
## Alberta            34.08697     18.75736       35.87457     3.085087
## Arizona            18.62427     26.62247       16.51613    30.872040
## Arkansas           31.46282     15.72412       26.21386     9.233346
## Armed Forces       30.94584     13.50146       27.68600     6.766233
## California         19.77606     30.89815       26.78642    38.940838
## Colorado           14.28653     25.19563       31.04362    32.568434
## Connecticut        23.65021     19.74667       25.74181    20.619079
## Delaware           30.70528     20.86518       33.05805    15.205185
## Florida            19.34592     22.28590       22.33636    32.843849
## Georgia            23.61336     17.27597       28.10998    21.122652
## Hawaii             33.26615     26.03658       31.55208    19.582621
## Idaho              25.08789     18.38610       25.90114    15.446663
## Illinois           26.78512     26.33174       24.07543    30.442896
## Indiana            17.79488     19.94497       21.71202    22.799632
## Iowa               27.57057     22.73166       25.05188    20.200136
## Kansas             28.75057     19.85495       30.69713    15.841424
## Kentucky           24.78457     17.94952       26.10844    14.964186
## Louisiana          26.21748     17.65026       16.61536    22.318592
## Maine              27.31149     21.57964       29.14053    11.835376
## Maryland           25.38533     30.97822       30.06218    36.504298
## Massachusetts      19.18344     21.01274       21.47217    29.948670
## Michigan           21.58466     18.68117       23.48297    25.228401
## Minnesota          25.84174     17.56240       18.23607    24.953889
## Mississippi        28.27276     14.57886       26.44877     9.638148
## Missouri           24.00963     16.32536       16.98473    22.212566
## Montana            28.01458     15.66401       27.27755     7.641202
## Nebraska           27.33394     20.05465       31.23489    20.681940
## Nevada             29.51528     16.50944       26.77426    11.809029
## New Hampshire      27.30102     16.58605       24.71646    11.254031
## New Jersey         19.58123     25.33159       24.79204    29.736893
## New Mexico         25.92925     19.20174       30.63842    14.265743
## New York           25.74294     27.30530       23.36747    35.292263
## North Carolina     20.77417     24.83108       20.28870    31.351864
## North Dakota       39.29046     29.96830       35.69142    14.296024
## Ohio               22.11209     19.91441       18.27370    24.418304
## Oklahoma           23.15760     19.60264       24.50275    18.536533
## Oregon             26.31792     14.62957       21.90069    20.165081
## Pennsylvania        0.00000     31.20766       25.39859    35.059742
## Rhode Island       31.20766      0.00000       30.55002    15.672278
## South Carolina     25.39859     30.55002        0.00000    32.789483
## South Dakota       35.05974     15.67228       32.78948     0.000000
## Tennessee          24.93321     17.31466       19.76710    28.000306
## Texas              18.33661     22.90491       25.36695    32.236074
## Utah               31.57173     24.27615       31.58471    24.297472
## Vermont            35.67005     15.37046       33.39980     2.490554
## Virginia           19.89497     18.54178       25.65974    23.821608
## Washington         22.05342     17.71249       28.03126    25.321358
## West Virginia      34.22984     18.07133       27.19259     7.105870
## Wisconsin          24.36934     21.67796       24.76939    20.205495
## Wyoming            29.39828     16.10200       29.36318    13.209772
## Grand Total        14.71537     28.56574       24.41957    37.013057
##                Tennessee    Texas     Utah   Vermont Virginia Washington
## Alabama         14.93753 22.00008 28.36697 14.909465 16.15828   19.72309
## Alaska          26.36773 27.32075 30.13563 19.459085 22.15049   26.50825
## Alberta         31.08539 33.43205 27.38256  4.066663 22.24550   28.40644
## Arizona         23.43490 17.12330 29.15756 30.320457 12.01769   23.44163
## Arkansas        19.55589 25.46400 26.36478  8.251770 20.43897   19.66753
## Armed Forces    22.61859 26.88626 19.70086  9.256787 18.16179   19.22893
## California      24.52261 16.96707 30.73134 37.959261 20.81706   24.10705
## Colorado        23.96986 14.88262 29.26040 31.586858 17.70627   20.01224
## Connecticut     22.19956 21.63475 20.35318 18.128524 18.30343   18.22826
## Delaware        27.67095 30.20784 23.16074 17.015983 21.79675   23.04260
## Florida         14.95354 16.06418 22.94255 31.862272 17.96680   16.26723
## Georgia         20.32039 19.56021 24.05124 21.341561 12.89627   16.54781
## Hawaii          28.08262 29.79658 31.32646 19.884441 23.95367   31.90381
## Idaho           19.21653 20.63783 24.56224 16.051755 15.88517   20.86914
## Illinois        21.23572 20.90209 29.11573 31.053209 24.62242   23.04593
## Indiana         18.36287 17.72862 15.06359 23.409945 17.58760   11.91217
## Iowa            22.50048 22.18526 16.42519 19.301471 17.33971   22.59049
## Kansas          28.25386 26.93913 23.31042 17.652222 17.85940   23.21125
## Kentucky        17.45635 21.60930 23.65585 15.574500 16.09293   19.99991
## Louisiana       18.54224 18.69466 24.39227 21.337016 21.36100   18.60865
## Maine           27.14500 24.59627 27.43905 13.646174 18.82310   25.60910
## Maryland        32.31315 21.74740 37.02819 37.114611 26.00000   28.23606
## Massachusetts   19.94648 19.87798 29.97592 28.967094 15.75056   20.61270
## Michigan        17.04879 19.90416 28.72772 24.246825 20.21054   17.97445
## Minnesota       12.99946 20.29458 26.70017 23.972312 19.16671   17.11091
## Mississippi     20.94355 26.38500 25.60408 10.248461 21.24415   20.88967
## Missouri        16.41083 20.48955 27.85171 21.230990 17.88002   20.95877
## Montana         20.88777 25.78378 20.19654  9.452000 19.17975   18.79448
## Nebraska        18.98002 25.74539 31.74879 20.900848 18.44795   23.40532
## Nevada          29.15476 24.84743 22.21335 13.619827 20.66282   23.19871
## New Hampshire   20.51203 24.16962 22.39836 10.355367 17.38603   20.14630
## New Jersey      19.08787 19.07810 24.68063 28.755317 15.83488   17.44250
## New Mexico      27.10566 22.09443 26.28197 14.484652 22.40265   24.86067
## New York        22.90558 19.24495 26.06734 34.310686 18.08918   21.65884
## North Carolina  22.12095 21.47042 25.94637 30.370288 19.20740   20.41562
## North Dakota    33.69589 34.95737 33.20933 14.597844 32.38027   31.69566
## Ohio            20.15045 14.92164 25.85799 24.637213 15.35873   20.08049
## Oklahoma        24.00527 18.64663 25.15568 20.347331 12.77366   24.21716
## Oregon          15.66301 22.40386 25.11153 19.183505 18.97065   18.30103
## Pennsylvania    24.93321 18.33661 31.57173 35.670055 19.89497   22.05342
## Rhode Island    17.31466 22.90491 24.27615 15.370458 18.54178   17.71249
## South Carolina  19.76710 25.36695 31.58471 33.399796 25.65974   28.03126
## South Dakota    28.00031 32.23607 24.29747  2.490554 23.82161   25.32136
## Tennessee        0.00000 19.96185 29.53419 27.018730 22.06429   16.93892
## Texas           19.96185  0.00000 29.33908 31.254498 17.60730   22.38514
## Utah            29.53419 29.33908  0.00000 24.907786 24.97453   18.16115
## Vermont         27.01873 31.25450 24.90779  0.000000 22.84003   24.33978
## Virginia        22.06429 17.60730 24.97453 22.840032  0.00000   20.14534
## Washington      16.93892 22.38514 18.16115 24.339781 20.14534    0.00000
## West Virginia   26.95640 31.53919 27.21704  7.407690 24.24968   24.40132
## Wisconsin       24.70430 18.84684 23.72038 20.815808 20.78618   22.03341
## Wyoming         21.78659 25.42904 20.49065 15.020570 22.65057   17.13217
## Grand Total     21.50378 14.32226 30.15433 36.031481 18.96896   20.38479
##                West Virginia Wisconsin  Wyoming Grand Total
## Alabama            13.356706  19.00884 14.82612   24.979412
## Alaska             20.939874  24.16372 17.59521   27.880298
## Alberta             8.681979  23.29058 16.29486   36.722376
## Arizona            26.670693  22.49466 28.29545   15.500243
## Arkansas           12.935840  16.91723 12.11137   29.868455
## Armed Forces       11.724147  18.36740 12.17348   31.835936
## California         37.968529  24.71531 31.29732    9.547551
## Colorado           32.203190  26.57896 26.53762   13.786153
## Connecticut        17.201120  22.80601 17.64555   20.243112
## Delaware           19.188065  22.52253 18.61169   31.041828
## Florida            31.945161  21.64806 24.87123   12.208366
## Georgia            19.526587  20.96400 18.44276   19.113110
## Hawaii             21.776137  23.98103 22.06156   32.220428
## Idaho              18.733029  16.83719 17.61650   26.009177
## Illinois           27.850578  19.10185 21.21878   19.072369
## Indiana            21.871689  18.66048 16.43106   17.527800
## Iowa               21.166618  18.30438 23.07958   25.229922
## Kansas             19.543918  21.06126 22.75935   30.231267
## Kentucky           17.469507  17.96518 17.83194   26.376060
## Louisiana          20.011711  19.10692 18.56922   21.326594
## Maine              15.290458  17.75196 18.06286   28.684144
## Maryland           33.811168  19.44736 31.05164   18.951180
## Massachusetts      27.142869  22.57782 24.10689   15.973438
## Michigan           21.454020  21.79468 21.03603   19.216696
## Minnesota          23.977188  19.33824 21.19326   20.561813
## Mississippi        11.356288  19.89572 10.40697   30.550270
## Missouri           22.411339  14.35388 18.92799   21.825512
## Montana            11.902261  18.37749 12.14699   31.204740
## Nebraska           23.246608  25.70756 26.38035   27.576315
## Nevada             15.511523  13.16067 14.48957   30.178811
## New Hampshire      11.787126  15.00491 16.12863   28.208958
## New Jersey         28.783912  24.67580 24.85597   14.351236
## New Mexico         17.757079  20.91936 19.24998   26.689173
## New York           29.695371  21.96888 25.45214   14.905399
## North Carolina     32.495496  24.12493 23.73809   19.066563
## North Dakota       18.637496  23.59596 20.31911   41.802868
## Ohio               21.157864  20.41626 23.30401   16.637702
## Oklahoma           18.456663  14.89053 20.32748   23.277135
## Oregon             20.348700  22.30860 22.65924   23.629231
## Pennsylvania       34.229843  24.36934 29.39828   14.715373
## Rhode Island       18.071329  21.67796 16.10200   28.565741
## South Carolina     27.192591  24.76939 29.36318   24.419573
## South Dakota        7.105870  20.20549 13.20977   37.013057
## Tennessee          26.956397  24.70430 21.78659   21.503781
## Texas              31.539185  18.84684 25.42904   14.322264
## Utah               27.217035  23.72038 20.49065   30.154331
## Vermont             7.407690  20.81581 15.02057   36.031481
## Virginia           24.249677  20.78618 22.65057   18.968962
## Washington         24.401322  22.03341 17.13217   20.384789
## West Virginia       0.000000  23.90799 14.35340   35.376629
## Wisconsin          23.907989   0.00000 17.58199   24.440180
## Wyoming            14.353404  17.58199  0.00000   29.403112
## Grand Total        35.376629  24.44018 29.40311    0.000000
clusterdata2<-hclust(dm1,method="average")
plot(clusterdata2, hang=-1, cex=0.7, main="Average Linkage Cluster")

#my next step is to see what are the optimum number of cluster that I can cut my data into for which I use Nbclust.

library(NbClust)
data2 <- NbClust(datascaled, distance="euclidean",min.nc=2, max.nc=12, method="average")

## *** : The Hubert index is a graphical method of determining the number of clusters.
##                 In the plot of Hubert index, we seek a significant knee that corresponds to a 
##                 significant increase of the value of the measure i.e the significant peak in Hubert
##                 index second differences plot. 
## 

## *** : The D index is a graphical method of determining the number of clusters. 
##                 In the plot of D index, we seek a significant knee (the significant peak in Dindex
##                 second differences plot) that corresponds to a significant increase of the value of
##                 the measure. 
##  
## ******************************************************************* 
## * Among all indices:                                                
## * 9 proposed 2 as the best number of clusters 
## * 2 proposed 3 as the best number of clusters 
## * 1 proposed 4 as the best number of clusters 
## * 2 proposed 5 as the best number of clusters 
## * 1 proposed 9 as the best number of clusters 
## * 2 proposed 10 as the best number of clusters 
## * 7 proposed 11 as the best number of clusters 
## 
##                    ***** Conclusion *****                            
##  
## * According to the majority rule, the best number of clusters is  2 
##  
##  
## *******************************************************************
table(data2$Best.n[1,])
## 
##  0  2  3  4  5  9 10 11 
##  2  9  2  1  2  1  2  7
barplot(table(data2$Best.n[1,]),xlab="Numer of Clusters", ylab="Number of Criteria",main="Number of Clusters Chosen by 26 Criteria")
#based on the plot, it appears 8 criteria show that 2 clusters would be the best way to cluster this data set. While 7 show that 11 clusters would be ideal. Using our descrition and knowledge of the data,it seems that 2 clusters would not be of much use. So I ve gone ahead and used 11 clusters in the next step.

clusters <- cutree(clusterdata, k=11)
table(clusters)
## clusters
##  1  2  3  4  5  6  7  8  9 10 11 
## 23 19  2  1  1  2  1  1  1  1  1
aggregate(data[,-1], by=list(cluster=clusters), mean)
##    cluster  Amethyst Aquamarine Black.Diamond Blue.Topaz   Citrine
## 1        1  750.1921  1072.0755      752.2279   690.3289  678.2364
## 2        2  458.5439   492.6009      205.6404   322.4263  178.4737
## 3        3  785.8403  1035.8346      756.4167   604.5714  400.2500
## 4        4  932.5000   553.0000        0.0000     0.0000    0.0000
## 5        5    0.0000     0.0000      410.0000     0.0000    0.0000
## 6        6 1375.6806   983.0625      695.4167   773.5833    0.0000
## 7        7  538.2500     0.0000      898.0000     0.0000    0.0000
## 8        8  325.0000     0.0000        0.0000     0.0000 1060.5000
## 9        9    0.0000     0.0000        0.0000     0.0000    0.0000
## 10      10  566.0000  1059.6000     1399.0000  1374.3333    0.0000
## 11      11 1239.0000   323.0000     1398.0000   326.0000    0.0000
##      Diamond  Emerald    Garnet  Morganite   Peridot Pink.Sapphire
## 1   339.6348 1358.637  650.9821  767.74235  502.6975      247.8528
## 2   218.0526 1029.699  205.1579   34.10526  143.7105      284.7368
## 3  4445.2500 1470.611  440.4333  741.00000  558.1000     1640.5000
## 4     0.0000  644.000    0.0000    0.00000    0.0000        0.0000
## 5     0.0000 2031.000    0.0000    0.00000 1400.0000     2512.0000
## 6  1738.0000 1251.493  957.7500    0.00000  706.2500      761.0000
## 7     0.0000 1413.833    0.0000    0.00000  439.0000        0.0000
## 8     0.0000  597.500  367.0000 1459.00000  650.0000        0.0000
## 9     0.0000 9962.000    0.0000    0.00000    0.0000        0.0000
## 10    0.0000 2135.250 2236.0000 2650.00000  569.5000      595.0000
## 11    0.0000  512.000    0.0000  777.50000    0.0000        0.0000
##    Pink.Tourmaline     Plain      Ruby  Sapphire SI.Diamond Tanzanite
## 1         683.0863  541.3469 1815.2820 1383.5890   810.7709 1015.8803
## 2         187.4737  135.5263  939.4718  942.3746   572.2982  653.8211
## 3        1028.0000  453.8500 1423.9048 2134.1500   894.1667 1214.6944
## 4           0.0000    0.0000    0.0000 1624.0000     0.0000 3588.0000
## 5           0.0000    0.0000 3375.0000 1802.7500     0.0000  332.0000
## 6         292.7500  419.6667 2087.0167 1382.6282  2208.5000 1037.2500
## 7           0.0000  457.0000 1210.6667 1592.0000     0.0000    0.0000
## 8        2191.0000  455.0000 1612.6000 3147.5000   589.0000  349.0000
## 9           0.0000    0.0000    0.0000    0.0000  2592.0000 1377.0000
## 10          0.0000  399.0000 3019.2500 1050.8333  1316.0000 1435.0000
## 11          0.0000 1011.0000 1048.5000 1160.0000     0.0000 1629.3333
##    VS.Diamond VVS.Diamond White.Sapphire White.Topaz Grand.Total
## 1    836.0638    561.8826      428.01732    334.4657   1151.2823
## 2    207.5263      0.0000       95.77368    153.5263    998.5589
## 3   2493.1000      0.0000      391.50000    475.7500   1179.3320
## 4   1338.0000   1355.0000      460.00000      0.0000   1204.8000
## 5      0.0000      0.0000        0.00000    698.0000   1531.7500
## 6    370.0000   4662.5000      683.50000    228.6667   1284.7682
## 7   7133.0000      0.0000        0.00000      0.0000   1542.7500
## 8      0.0000      0.0000        0.00000      0.0000   1187.0000
## 9      0.0000      0.0000        0.00000      0.0000   4643.6667
## 10     0.0000      0.0000      473.00000      0.0000   1412.5778
## 11  1495.3333      0.0000        0.00000   2514.0000   1100.9600
aggregate(data[,-1], by=list(cluster=clusters), median)
##    cluster  Amethyst Aquamarine Black.Diamond Blue.Topaz   Citrine Diamond
## 1        1  742.3333   994.3333      826.0000   575.0000  666.3333    0.00
## 2        2  388.0000   406.0000        0.0000     0.0000    0.0000    0.00
## 3        3  785.8403  1035.8346      756.4167   604.5714  400.2500 4445.25
## 4        4  932.5000   553.0000        0.0000     0.0000    0.0000    0.00
## 5        5    0.0000     0.0000      410.0000     0.0000    0.0000    0.00
## 6        6 1375.6806   983.0625      695.4167   773.5833    0.0000 1738.00
## 7        7  538.2500     0.0000      898.0000     0.0000    0.0000    0.00
## 8        8  325.0000     0.0000        0.0000     0.0000 1060.5000    0.00
## 9        9    0.0000     0.0000        0.0000     0.0000    0.0000    0.00
## 10      10  566.0000  1059.6000     1399.0000  1374.3333    0.0000    0.00
## 11      11 1239.0000   323.0000     1398.0000   326.0000    0.0000    0.00
##     Emerald    Garnet Morganite Peridot Pink.Sapphire Pink.Tourmaline
## 1  1388.143  536.3333     868.0  545.00           0.0          635.00
## 2  1060.000    0.0000       0.0    0.00           0.0            0.00
## 3  1470.611  440.4333     741.0  558.10        1640.5         1028.00
## 4   644.000    0.0000       0.0    0.00           0.0            0.00
## 5  2031.000    0.0000       0.0 1400.00        2512.0            0.00
## 6  1251.493  957.7500       0.0  706.25         761.0          292.75
## 7  1413.833    0.0000       0.0  439.00           0.0            0.00
## 8   597.500  367.0000    1459.0  650.00           0.0         2191.00
## 9  9962.000    0.0000       0.0    0.00           0.0            0.00
## 10 2135.250 2236.0000    2650.0  569.50         595.0            0.00
## 11  512.000    0.0000     777.5    0.00           0.0            0.00
##        Plain     Ruby Sapphire SI.Diamond Tanzanite VS.Diamond VVS.Diamond
## 1   575.1685 1721.979 1447.000   801.7500  1100.945    768.000         0.0
## 2     0.0000  565.000  985.000     0.0000   680.000      0.000         0.0
## 3   453.8500 1423.905 2134.150   894.1667  1214.694   2493.100         0.0
## 4     0.0000    0.000 1624.000     0.0000  3588.000   1338.000      1355.0
## 5     0.0000 3375.000 1802.750     0.0000   332.000      0.000         0.0
## 6   419.6667 2087.017 1382.628  2208.5000  1037.250    370.000      4662.5
## 7   457.0000 1210.667 1592.000     0.0000     0.000   7133.000         0.0
## 8   455.0000 1612.600 3147.500   589.0000   349.000      0.000         0.0
## 9     0.0000    0.000    0.000  2592.0000  1377.000      0.000         0.0
## 10  399.0000 3019.250 1050.833  1316.0000  1435.000      0.000         0.0
## 11 1011.0000 1048.500 1160.000     0.0000  1629.333   1495.333         0.0
##    White.Sapphire White.Topaz Grand.Total
## 1           495.0    402.0000    1154.123
## 2             0.0      0.0000    1051.143
## 3           391.5    475.7500    1179.332
## 4           460.0      0.0000    1204.800
## 5             0.0    698.0000    1531.750
## 6           683.5    228.6667    1284.768
## 7             0.0      0.0000    1542.750
## 8             0.0      0.0000    1187.000
## 9             0.0      0.0000    4643.667
## 10          473.0      0.0000    1412.578
## 11            0.0   2514.0000    1100.960
plot(clusterdata, hang=-1, cex=.8,main="Average Linkage Clustering\n11 Cluster Solution")
rect.hclust(clusterdata, k=11)

c1 <- data.frame(data$State,clusters)
a<-c1$data.State[c1$clusters==1]
b<-c1$data.State[c1$clusters==2]

#we use 11 clusters to cluster the data. We also view the mean and median price of each cluster by gemstone.
# from the new dendogram it appears there are 2 main clusters we can use and further analyze. One constitutes of 25 states and 1 of 16. we view the states in these 2 clusters. It appears that the median for the 1st cluster is higher for several stones like amethyst, aquamarine, sapphire, ruby vs the 3rd cluster. Maybe we can investigate if these states are likely to spend more on fine jewelry vs the states in cluster 3.

clusters2 <- cutree(clusterdata2, k=11)
table(clusters2)
## clusters2
##  1  2  3  4  5  6  7  8  9 10 11 
## 11 20 10  3  1  2  2  1  1  1  1
aggregate(data, by=list(cluster=clusters2), mean)
## Warning in mean.default(X[[i]], ...): argument is not numeric or logical:
## returning NA

## Warning in mean.default(X[[i]], ...): argument is not numeric or logical:
## returning NA

## Warning in mean.default(X[[i]], ...): argument is not numeric or logical:
## returning NA

## Warning in mean.default(X[[i]], ...): argument is not numeric or logical:
## returning NA

## Warning in mean.default(X[[i]], ...): argument is not numeric or logical:
## returning NA

## Warning in mean.default(X[[i]], ...): argument is not numeric or logical:
## returning NA

## Warning in mean.default(X[[i]], ...): argument is not numeric or logical:
## returning NA

## Warning in mean.default(X[[i]], ...): argument is not numeric or logical:
## returning NA

## Warning in mean.default(X[[i]], ...): argument is not numeric or logical:
## returning NA

## Warning in mean.default(X[[i]], ...): argument is not numeric or logical:
## returning NA

## Warning in mean.default(X[[i]], ...): argument is not numeric or logical:
## returning NA
##    cluster State  Amethyst Aquamarine Black.Diamond Blue.Topaz   Citrine
## 1        1    NA  709.7091  1176.5611      413.4091   933.6424  819.0455
## 2        2    NA  527.4917   430.5833      213.4333   221.2850   91.1500
## 3        3    NA  786.7269   986.6637      900.3242   576.6023  640.9509
## 4        4    NA  727.2269   980.8291      861.9444   585.3393  674.1667
## 5        5    NA    0.0000     0.0000      410.0000     0.0000    0.0000
## 6        6    NA 1375.6806   983.0625      695.4167   773.5833    0.0000
## 7        7    NA  619.5000   322.5000     1544.0000   440.8333    0.0000
## 8        8    NA  325.0000     0.0000        0.0000     0.0000 1060.5000
## 9        9    NA  603.6000  1956.8571     1524.0000   439.3333  526.4286
## 10      10    NA    0.0000     0.0000        0.0000     0.0000    0.0000
## 11      11    NA  566.0000  1059.6000     1399.0000  1374.3333    0.0000
##       Diamond  Emerald    Garnet Morganite   Peridot Pink.Sapphire
## 1    59.72727 1084.004  602.1970  746.4545  199.5000      218.3636
## 2   207.15000 1048.989  166.0750    0.0000  221.7750      150.4000
## 3   424.26000 1597.024  845.9365  718.6074  661.3041      513.1615
## 4  3934.16667 1560.593  448.8074  494.0000  581.5667     1093.6667
## 5     0.00000 2031.000    0.0000    0.0000 1400.0000     2512.0000
## 6  1738.00000 1251.493  957.7500    0.0000  706.2500      761.0000
## 7     0.00000  690.000    0.0000  388.7500  272.5000        0.0000
## 8     0.00000  597.500  367.0000 1459.0000  650.0000        0.0000
## 9     0.00000 1388.143    0.0000 2909.0000  315.0000      569.0000
## 10    0.00000 9962.000    0.0000    0.0000    0.0000        0.0000
## 11    0.00000 2135.250 2236.0000 2650.0000  569.5000      595.0000
##    Pink.Tourmaline    Plain     Ruby Sapphire SI.Diamond Tanzanite
## 1         176.1364 356.3939 2086.339 1284.579   682.6061  942.5091
## 2         191.3500 141.2000  805.219 1010.462   543.6833  734.9500
## 3        1180.8651 697.7288 1679.888 1441.845   846.1690 1054.9600
## 4         870.2778 457.8524 1651.193 1991.119  1219.4028 1203.7261
## 5           0.0000   0.0000 3375.000 1802.750     0.0000  332.0000
## 6         292.7500 419.6667 2087.017 1382.628  2208.5000 1037.2500
## 7         236.0000 964.7500 1069.375 1189.143     0.0000 1648.6667
## 8        2191.0000 455.0000 1612.600 3147.500   589.0000  349.0000
## 9         673.0000 377.0000 1763.100 1262.250   807.5000  909.8571
## 10          0.0000   0.0000    0.000    0.000  2592.0000 1377.0000
## 11          0.0000 399.0000 3019.250 1050.833  1316.0000 1435.0000
##    VS.Diamond VVS.Diamond White.Sapphire White.Topaz Grand.Total
## 1    693.9091    89.36364       124.2424    257.1818    1105.631
## 2    492.2000    67.75000       154.6350    103.8500    1040.743
## 3   1031.7467  1110.83000       595.6732    384.6711    1186.533
## 4   1918.0667     0.00000       559.3333    586.1667    1217.014
## 5      0.0000     0.00000         0.0000    698.0000    1531.750
## 6    370.0000  4662.50000       683.5000    228.6667    1284.768
## 7    747.6667   416.00000       129.5000   1458.0000    1005.998
## 8      0.0000     0.00000         0.0000      0.0000    1187.000
## 9   3081.0000     0.00000       554.0000    648.0000    1154.123
## 10     0.0000     0.00000         0.0000      0.0000    4643.667
## 11     0.0000     0.00000       473.0000      0.0000    1412.578
aggregate(data[,-1], by=list(cluster=clusters2), median)
##    cluster  Amethyst Aquamarine Black.Diamond Blue.Topaz   Citrine Diamond
## 1        1  684.5000  1202.1250      471.0000   876.0000  741.0000       0
## 2        2  480.1250   338.0000        0.0000     0.0000    0.0000       0
## 3        3  796.4786   932.5694      954.8333   556.6000  647.8333       0
## 4        4  761.1250   972.3750      998.5000   563.1429  800.5000    3752
## 5        5    0.0000     0.0000      410.0000     0.0000    0.0000       0
## 6        6 1375.6806   983.0625      695.4167   773.5833    0.0000    1738
## 7        7  619.5000   322.5000     1544.0000   440.8333    0.0000       0
## 8        8  325.0000     0.0000        0.0000     0.0000 1060.5000       0
## 9        9  603.6000  1956.8571     1524.0000   439.3333  526.4286       0
## 10      10    0.0000     0.0000        0.0000     0.0000    0.0000       0
## 11      11  566.0000  1059.6000     1399.0000  1374.3333    0.0000       0
##     Emerald    Garnet Morganite   Peridot Pink.Sapphire Pink.Tourmaline
## 1  1080.143  536.3333    881.00    0.0000             0           0.000
## 2  1083.917    0.0000      0.00    0.0000             0           0.000
## 3  1484.920  698.1049    743.00  699.0152             0        1172.234
## 4  1728.222  465.5556      0.00  628.5000          1150         864.200
## 5  2031.000    0.0000      0.00 1400.0000          2512           0.000
## 6  1251.493  957.7500      0.00  706.2500           761         292.750
## 7   690.000    0.0000    388.75  272.5000             0         236.000
## 8   597.500  367.0000   1459.00  650.0000             0        2191.000
## 9  1388.143    0.0000   2909.00  315.0000           569         673.000
## 10 9962.000    0.0000      0.00    0.0000             0           0.000
## 11 2135.250 2236.0000   2650.00  569.5000           595           0.000
##       Plain     Ruby Sapphire SI.Diamond Tanzanite VS.Diamond VVS.Diamond
## 1  310.0000 2049.500 1338.818    732.500 1000.0000     0.0000         0.0
## 2    0.0000  468.500  987.500      0.000  643.0000     0.0000         0.0
## 3  658.9762 1676.633 1524.663   1005.500 1104.8475  1269.5000       652.5
## 4  456.5000 1434.667 2109.300   1075.333 1181.7895  2425.2000         0.0
## 5    0.0000 3375.000 1802.750      0.000  332.0000     0.0000         0.0
## 6  419.6667 2087.017 1382.628   2208.500 1037.2500   370.0000      4662.5
## 7  964.7500 1069.375 1189.143      0.000 1648.6667   747.6667       416.0
## 8  455.0000 1612.600 3147.500    589.000  349.0000     0.0000         0.0
## 9  377.0000 1763.100 1262.250    807.500  909.8571  3081.0000         0.0
## 10   0.0000    0.000    0.000   2592.000 1377.0000     0.0000         0.0
## 11 399.0000 3019.250 1050.833   1316.000 1435.0000     0.0000         0.0
##    White.Sapphire White.Topaz Grand.Total
## 1             0.0      0.0000    1050.000
## 2             0.0      0.0000    1096.424
## 3           646.6    519.4167    1191.793
## 4           783.0    574.5000    1283.598
## 5             0.0    698.0000    1531.750
## 6           683.5    228.6667    1284.768
## 7           129.5   1458.0000    1005.998
## 8             0.0      0.0000    1187.000
## 9           554.0    648.0000    1154.123
## 10            0.0      0.0000    4643.667
## 11          473.0      0.0000    1412.578
plot(clusterdata2, hang=-1, cex=.8,main="Average Linkage Clustering\n11 Cluster Solution")
rect.hclust(clusterdata2, k=11)

c2 <- data.frame(data$State,clusters2)
a2<-c2$data.State[c1$clusters2==1]
b2<-c2$data.State[c1$clusters2==2]

I ve also run the partioning around medoid cluster on the same dataset, this time using 2 clusters.

library(cluster)
set.seed(789)
data<-data[1:52,]
rownames(data) <- data$State
pamdata <- pam(data[-1], k=2, stand=TRUE)
pamdata$medoids
##         Amethyst Aquamarine Black.Diamond Blue.Topaz Citrine Diamond
## Ohio      677.75      763.8        1323.5          0     862     433
## Montana   533.25      480.0           0.0          0       0       0
##          Emerald Garnet Morganite Peridot Pink.Sapphire Pink.Tourmaline
## Ohio    1826.412   1114       790   586.5             0             635
## Montana  644.000      0         0     0.0             0               0
##          Plain     Ruby Sapphire SI.Diamond Tanzanite VS.Diamond
## Ohio    530.75 1512.833 1505.778       1029   622.125          0
## Montana   0.00    0.000 1675.000        482  1539.500          0
##         VVS.Diamond White.Sapphire White.Topaz Grand.Total
## Ohio              0            495           0    1118.299
## Montana           0              0           0     849.300
clusplot(pamdata, main="Cluster Plot")

ct.pam <- table(data$State, pamdata$clustering)
summary(pamdata)
## Medoids:
##         ID Amethyst Aquamarine Black.Diamond Blue.Topaz Citrine Diamond
## Ohio    37   677.75      763.8        1323.5          0     862     433
## Montana 28   533.25      480.0           0.0          0       0       0
##          Emerald Garnet Morganite Peridot Pink.Sapphire Pink.Tourmaline
## Ohio    1826.412   1114       790   586.5             0             635
## Montana  644.000      0         0     0.0             0               0
##          Plain     Ruby Sapphire SI.Diamond Tanzanite VS.Diamond
## Ohio    530.75 1512.833 1505.778       1029   622.125          0
## Montana   0.00    0.000 1675.000        482  1539.500          0
##         VVS.Diamond White.Sapphire White.Topaz Grand.Total
## Ohio              0            495           0    1118.299
## Montana           0              0           0     849.300
## Clustering vector:
##        Alabama         Alaska        Alberta        Arizona       Arkansas 
##              1              2              2              1              2 
##   Armed Forces     California       Colorado    Connecticut       Delaware 
##              2              1              1              2              2 
##        Florida        Georgia         Hawaii          Idaho       Illinois 
##              1              1              1              2              1 
##        Indiana           Iowa         Kansas       Kentucky      Louisiana 
##              1              1              1              2              1 
##          Maine       Maryland  Massachusetts       Michigan      Minnesota 
##              2              1              1              1              1 
##    Mississippi       Missouri        Montana       Nebraska         Nevada 
##              2              1              2              1              1 
##  New Hampshire     New Jersey     New Mexico       New York North Carolina 
##              2              1              2              1              1 
##   North Dakota           Ohio       Oklahoma         Oregon   Pennsylvania 
##              1              1              1              1              1 
##   Rhode Island South Carolina   South Dakota      Tennessee          Texas 
##              1              1              2              1              1 
##           Utah        Vermont       Virginia     Washington  West Virginia 
##              2              2              1              1              2 
##      Wisconsin        Wyoming 
##              1              2 
## Objective function:
##    build     swap 
## 6.477915 6.438597 
## 
## Numerical information per cluster:
##      size max_diss  av_diss diameter separation
## [1,]   34 21.73284 6.858750 23.75496   4.634538
## [2,]   18  9.93505 5.644976 11.81608   4.634538
## 
## Isolated clusters:
##  L-clusters: character(0)
##  L*-clusters: character(0)
## 
## Silhouette plot information:
##                cluster neighbor     sil_width
## Florida              1        2  0.1503071132
## California           1        2  0.1437597228
## New York             1        2  0.1383340516
## Arizona              1        2  0.1319183678
## South Carolina       1        2  0.1272512494
## Massachusetts        1        2  0.1196627294
## Maryland             1        2  0.1063676787
## Texas                1        2  0.1023169331
## North Carolina       1        2  0.0860813673
## Tennessee            1        2  0.0846385910
## New Jersey           1        2  0.0821861397
## Pennsylvania         1        2  0.0711451448
## Ohio                 1        2  0.0652021086
## Illinois             1        2  0.0594267018
## Minnesota            1        2  0.0465182555
## Colorado             1        2  0.0394459137
## Missouri             1        2  0.0370280696
## Virginia             1        2  0.0356382673
## Michigan             1        2  0.0232531280
## Louisiana            1        2  0.0028592824
## North Dakota         1        2  0.0009838071
## Washington           1        2 -0.0003758998
## Nebraska             1        2 -0.0228369100
## Oregon               1        2 -0.0380125358
## Georgia              1        2 -0.0405276658
## Indiana              1        2 -0.0519177327
## Hawaii               1        2 -0.0531651135
## Oklahoma             1        2 -0.0558880262
## Kansas               1        2 -0.0565971103
## Wisconsin            1        2 -0.0592056713
## Rhode Island         1        2 -0.0749345491
## Alabama              1        2 -0.1103860744
## Iowa                 1        2 -0.1146202310
## Nevada               1        2 -0.1389059595
## South Dakota         2        1  0.3229943523
## Vermont              2        1  0.3184603233
## Montana              2        1  0.3102890220
## Alberta              2        1  0.2687780632
## Mississippi          2        1  0.2438581841
## Armed Forces         2        1  0.2406387304
## West Virginia        2        1  0.2229990570
## Arkansas             2        1  0.2090135485
## New Hampshire        2        1  0.2046689836
## Maine                2        1  0.1663806191
## Wyoming              2        1  0.1536789472
## Idaho                2        1  0.1469198952
## Kentucky             2        1  0.1391266121
## Delaware             2        1  0.1316644826
## Alaska               2        1  0.1239947645
## New Mexico           2        1  0.1208823242
## Connecticut          2        1  0.1148016579
## Utah                 2        1  0.0263036850
## Average silhouette width per cluster:
## [1] 0.02461621 0.19252518
## Average silhouette width of total data set:
## [1] 0.08273855
## 
## 1326 dissimilarities, summarized :
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  1.6227  6.9616  8.2351  8.8040  9.7861 25.3270 
## Metric :  euclidean 
## Number of objects : 52
## 
## Available components:
##  [1] "medoids"    "id.med"     "clustering" "objective"  "isolation" 
##  [6] "clusinfo"   "silinfo"    "diss"       "call"       "data"
pamdataframe <- as.data.frame(pamdata$clustering)
c3 <- data.frame(data$State[-53],pamdataframe)
a3<-c3$data.State..53.[c3$pamdata.clustering==1]
b3<-c3$data.State..53.[c3$pamdata.clustering==2]

# I ve broken the data into 2 clusters, the 2 medoid are Ohio and Montana. The higher avg price states are placed in the ohio medoid. COmparing the 2 results, we see a lot of similarities. California, Ohio, Florida, Georgia, arizona etc are all in cluster 1 (for the medoids methoda as well as cutree.) Hence, these states most likely have a higer average price across gemstones and we can target our customers accordingly.
highspendingstates<-intersect(a,a3)
lowspendingstates<-intersect(b,b3)
highspendingstates
##  [1] "Alabama"        "Arizona"        "California"     "Florida"       
##  [5] "Georgia"        "Indiana"        "Iowa"           "Louisiana"     
##  [9] "Massachusetts"  "Michigan"       "Minnesota"      "Missouri"      
## [13] "New Jersey"     "New York"       "North Carolina" "Ohio"          
## [17] "Oklahoma"       "Rhode Island"   "Tennessee"      "Texas"         
## [21] "Virginia"       "Washington"
lowspendingstates
##  [1] "Alaska"        "Alberta"       "Arkansas"      "Armed Forces" 
##  [5] "Connecticut"   "Idaho"         "Kentucky"      "Maine"        
##  [9] "Mississippi"   "Montana"       "New Hampshire" "New Mexico"   
## [13] "South Dakota"  "Vermont"       "West Virginia" "Wyoming"
hi<-as.matrix(clusters)
pamcv<-as.matrix(pamdata$clustering)

#trying to validate clusters.
library(clValid)
hi.cv<-clValid(hi,11,clMethods = c("hierarchical"),validation="internal")
pamcv.cv<-clValid(pamcv,2,clMethods = c("pam"),validation="internal")
summary(hi.cv)
## 
## Clustering Methods:
##  hierarchical 
## 
## Cluster sizes:
##  11 
## 
## Validation Measures:
##                                 11
##                                   
## hierarchical Connectivity  28.2187
##              Dunn              Inf
##              Silhouette     0.8679
## 
## Optimal Scores:
## 
##              Score   Method       Clusters
## Connectivity 28.2187 hierarchical 11      
## Dunn             Inf hierarchical 11      
## Silhouette    0.8679 hierarchical 11
summary(pamcv.cv)
## 
## Clustering Methods:
##  pam 
## 
## Cluster sizes:
##  2 
## 
## Validation Measures:
##                     2
##                      
## pam Connectivity    0
##     Dunn          Inf
##     Silhouette      1
## 
## Optimal Scores:
## 
##              Score Method Clusters
## Connectivity   0   pam    2       
## Dunn         Inf   pam    2       
## Silhouette     1   pam    2
#connectivity is lower for Pam and silhouette is higher for Pam, so it appears to be the better form of clustering.
#Connectivity - what extent items are placed in the same cluster as their nearest neighbors in the data space. It has a value between 0 and infinity and should be minimized.
#Average Silhouette width - It lies between -1 (poorly clustered observations) to 1 (well clustered observations). It should be maximized.

ANISH ———————————————————————

Using PAM for clustering states by sales

library(xlsx)
## Loading required package: rJava
## Loading required package: xlsxjars
library(plyr)
## -------------------------------------------------------------------------
## You have loaded plyr after dplyr - this is likely to cause problems.
## If you need functions from both plyr and dplyr, please load plyr first, then dplyr:
## library(plyr); library(dplyr)
## -------------------------------------------------------------------------
## 
## Attaching package: 'plyr'
## The following objects are masked from 'package:dplyr':
## 
##     arrange, count, desc, failwith, id, mutate, rename, summarise,
##     summarize
## The following object is masked from 'package:purrr':
## 
##     compact
## The following objects are masked from 'package:Hmisc':
## 
##     is.discrete, summarize
library(dplyr)
library(cluster)
library(flexclust)
## Loading required package: grid
## Loading required package: modeltools
## Loading required package: stats4
## 
## Attaching package: 'modeltools'
## The following object is masked from 'package:plyr':
## 
##     empty
## The following object is masked from 'package:rJava':
## 
##     clone
## The following object is masked from 'package:clValid':
## 
##     clusters
x <- read.xlsx("Files/Anish/Sales cluster dataset.xlsx",2)
x[is.na(x)] <- 0
glimpse(x)
## Observations: 51
## Variables: 8
## $ State     <fctr> Alabama, Alaska, Arizona, Arkansas, Armed Forces, C...
## $ Sunday    <dbl> 6, 0, 2, 0, 0, 5, 0, 2, 2, 3, 3, 1, 1, 5, 1, 5, 1, 4...
## $ Monday    <dbl> 3, 2, 7, 4, 2, 9, 3, 3, 2, 4, 5, 3, 3, 0, 3, 4, 3, 6...
## $ Tuesday   <dbl> 4, 1, 9, 6, 0, 4, 5, 2, 4, 7, 4, 5, 7, 6, 3, 12, 2, ...
## $ Wednesday <dbl> 7, 4, 9, 1, 2, 6, 1, 4, 3, 6, 7, 2, 0, 1, 1, 5, 5, 3...
## $ Thursday  <dbl> 6, 3, 9, 2, 0, 7, 0, 1, 2, 8, 4, 3, 6, 7, 3, 9, 3, 3...
## $ Friday    <dbl> 5, 2, 7, 1, 0, 4, 1, 1, 3, 13, 5, 5, 8, 3, 2, 7, 2, ...
## $ Saturday  <dbl> 5, 3, 8, 1, 1, 2, 0, 3, 1, 2, 0, 1, 4, 7, 2, 5, 0, 2...
rownames(x) <- x$State

#The data needs to be scaled to run the clustering algorithm
scaled <- data.frame(scale(x[,2:8]))
scaledx <- data.frame(scaled)
rownames(scaledx) <- x$State


#A distance formula needs to be run to calculate distances on the scaled data frame
distx <- dist(scaledx,method = "euclidean")
#This is the average linkage clustering
fit.average <- hclust(distx, method="average")
plot(fit.average, hang=-1, cex=.8, main="Average Linkage Clustering")

#Let's split into 3 clusters
clusters <- cutree(fit.average, k=3)
table(clusters)
## clusters
##  1  2  3 
## 47  1  3
#These are the average mean and medians of the clusters
#Looks like cluster 1 has low number of sales, cluster 2 has the highest and cluster 3 has the second highest. Clusters 2 and 3 means are close to each other compared to cluster 1 but the difference is a bit more pronounced in the medians. 
aggregate(x[,-1], by=list(cluster=clusters), mean)
##   cluster    Sunday   Monday   Tuesday Wednesday Thursday    Friday
## 1       1  3.914894  5.93617  6.212766         6  6.12766  6.446809
## 2       2 25.000000 45.00000 37.000000        51 55.00000 53.000000
## 3       3 22.000000 28.66667 30.000000        34 27.33333 29.000000
##   Saturday
## 1  4.00000
## 2 51.00000
## 3 23.33333
aggregate(x[,-1], by=list(cluster=clusters), median)
##   cluster Sunday Monday Tuesday Wednesday Thursday Friday Saturday
## 1       1      3      4       5         4        6      4        3
## 2       2     25     45      37        51       55     53       51
## 3       3     19     28      31        35       29     29       27
plot(fit.average, hang=-1, cex=.8,
       main="Average Linkage Clustering\n3 Cluster Solution")

#It has split up CA into its own cluster, NY, FL & TX into their own cluster as the second best performing and all the other states into one big cluster of their own
rect.hclust(fit.average, k=3)

plot(silhouette(clusters, distx), main = "HClust Silhouette Plot")

#Let's see the results by running another clustering algorithm - Partitioning around medoids
pamcluster <- pam(distx, 3, diss=TRUE, stand=TRUE)

#Looks like this algorithm has grouped the states differently from the earlier algorithm. 
# The top 4 states CA, NY, TX, and FL have been put into their own cluster followed by the next 13 states in the second cluster followed by the rest into the 3rd cluster. This seems like a better result. 
summary(pamcluster)
## Medoids:
##      ID             
## [1,] "25" "Oklahoma"
## [2,] "38" "Texas"   
## [3,] "47" "Ohio"    
## Clustering vector:
##        Alabama         Alaska        Arizona       Arkansas   Armed Forces 
##              1              1              1              1              1 
##    Connecticut       Delaware         Hawaii          Idaho        Indiana 
##              1              1              1              1              1 
##           Iowa         Kansas       Kentucky      Louisiana          Maine 
##              1              1              1              1              1 
##      Minnesota    Mississippi       Missouri        Montana       Nebraska 
##              1              1              1              1              1 
##         Nevada  New Hampshire     New Mexico   North Dakota       Oklahoma 
##              1              1              1              1              1 
##         Oregon   Rhode Island South Carolina   South Dakota           Utah 
##              1              1              1              1              1 
##        Vermont  West Virginia      Wisconsin        Wyoming     California 
##              1              1              1              1              2 
##        Florida       New York          Texas       Colorado        Georgia 
##              2              2              2              3              3 
##       Illinois       Maryland  Massachusetts       Michigan     New Jersey 
##              3              3              3              3              3 
## North Carolina           Ohio   Pennsylvania      Tennessee       Virginia 
##              3              3              3              3              3 
##     Washington 
##              3 
## Objective function:
##     build      swap 
## 0.9740851 0.9521850 
## 
## Numerical information per cluster:
##      size max_diss   av_diss diameter separation
## [1,]   34 1.354657 0.6832387 2.114534  0.9363351
## [2,]    4 4.967819 2.5835405 6.257107  3.7055411
## [3,]   13 1.788970 1.1536273 2.407475  0.9363351
## 
## Isolated clusters:
##  L-clusters: character(0)
##  L*-clusters: character(0)
## 
## Silhouette plot information:
##                cluster neighbor   sil_width
## Maine                1        3  0.75579229
## Idaho                1        3  0.74725942
## Oklahoma             1        3  0.74722865
## Wyoming              1        3  0.74164774
## New Mexico           1        3  0.73701700
## Hawaii               1        3  0.73495325
## Mississippi          1        3  0.73139719
## Alaska               1        3  0.72711654
## West Virginia        1        3  0.72549808
## Montana              1        3  0.72500073
## Kansas               1        3  0.72179314
## Armed Forces         1        3  0.72009023
## Rhode Island         1        3  0.71783910
## Vermont              1        3  0.71365923
## Delaware             1        3  0.71071113
## North Dakota         1        3  0.71040391
## Nevada               1        3  0.70685791
## South Dakota         1        3  0.70517227
## Arkansas             1        3  0.70479485
## Nebraska             1        3  0.69977038
## New Hampshire        1        3  0.69951552
## Utah                 1        3  0.65967874
## Iowa                 1        3  0.63989827
## Kentucky             1        3  0.59416727
## Missouri             1        3  0.59246419
## Louisiana            1        3  0.54440144
## Wisconsin            1        3  0.52953656
## Alabama              1        3  0.51219590
## Connecticut          1        3  0.47290892
## Oregon               1        3  0.43841043
## Indiana              1        3  0.35776295
## South Carolina       1        3  0.35209659
## Minnesota            1        3  0.22171289
## Arizona              1        3  0.20380224
## Texas                2        3  0.42825283
## California           2        3  0.42488334
## Florida              2        3  0.20099615
## New York             2        3  0.12759960
## Pennsylvania         3        1  0.59566169
## Illinois             3        1  0.56912339
## Ohio                 3        1  0.56764758
## New Jersey           3        1  0.56303216
## Virginia             3        1  0.55958231
## Colorado             3        1  0.52409034
## North Carolina       3        1  0.51606267
## Michigan             3        1  0.39816893
## Maryland             3        1  0.38508030
## Washington           3        1  0.35056776
## Massachusetts        3        1  0.18693317
## Georgia              3        1  0.18298426
## Tennessee            3        1 -0.02814095
## Average silhouette width per cluster:
## [1] 0.6265457 0.2954330 0.4131380
## Average silhouette width of total data set:
## [1] 0.546178
## 
## Available components:
## [1] "medoids"    "id.med"     "clustering" "objective"  "isolation" 
## [6] "clusinfo"   "silinfo"    "diss"       "call"
clusplot(pamcluster, main="Bivariate Cluster Plot")

pamcluster$medoids
## [1] "Oklahoma" "Texas"    "Ohio"
plot(silhouette(pamcluster, distx), main = "PAM Silhouette Plot")

d <- as.data.frame(pamcluster$clustering)
pamdataset <- data.frame(x,d)
aggregate(pamdataset[,-1], by=list(pamdataset[,9]), mean)
##   Group.1    Sunday    Monday   Tuesday Wednesday  Thursday    Friday
## 1       1  2.147059  3.264706  3.794118  3.088235  3.705882  3.411765
## 2       2 22.750000 32.750000 31.750000 38.250000 34.250000 35.000000
## 3       3  8.538462 12.923077 12.538462 13.615385 12.461538 14.384615
##    Saturday pamcluster.clustering
## 1  2.441176                     1
## 2 30.250000                     2
## 3  8.076923                     3
aggregate(pamdataset[,-1], by=list(pamdataset[,9]), median)
##   Group.1 Sunday Monday Tuesday Wednesday Thursday Friday Saturday
## 1       1      2    3.0       4         3      3.0    2.0        2
## 2       2     22   29.5      32        35     31.5   34.5       27
## 3       3     10   14.0      12        15     13.0   14.0        8
##   pamcluster.clustering
## 1                     1
## 2                     2
## 3                     3
#Average linkage clustering shows CA as the top performing state followed by NY, FL, & TX but then combines everything else together. 
#PAM clustering seems to have split the results into a better cluster of 
#Top states = CA, NY, TX, FL 
#Middle States = CO, GA, IL, MD, MA, MI, NJ, NC, OH, PA, TN, VA, WA
#Low performing = The rest.
# The reason I prefer PAM for this dataset is because though both select the same 4 states as the top performers, PAM gives more insight into the middle and low performing states.
library(NbClust)

nc <- NbClust(scaledx,distance = "euclidean",min.nc = 2, max.nc = 15,method = "average")
## Warning in pf(beale, pp, df2): NaNs produced

## Warning in pf(beale, pp, df2): NaNs produced
## [1] "Frey index : No clustering structure in this data set"

## *** : The Hubert index is a graphical method of determining the number of clusters.
##                 In the plot of Hubert index, we seek a significant knee that corresponds to a 
##                 significant increase of the value of the measure i.e the significant peak in Hubert
##                 index second differences plot. 
## 

## *** : The D index is a graphical method of determining the number of clusters. 
##                 In the plot of D index, we seek a significant knee (the significant peak in Dindex
##                 second differences plot) that corresponds to a significant increase of the value of
##                 the measure. 
##  
## ******************************************************************* 
## * Among all indices:                                                
## * 6 proposed 2 as the best number of clusters 
## * 5 proposed 3 as the best number of clusters 
## * 2 proposed 4 as the best number of clusters 
## * 4 proposed 5 as the best number of clusters 
## * 1 proposed 6 as the best number of clusters 
## * 4 proposed 10 as the best number of clusters 
## * 1 proposed 14 as the best number of clusters 
## 
##                    ***** Conclusion *****                            
##  
## * According to the majority rule, the best number of clusters is  2 
##  
##  
## *******************************************************************
table(nc$Best.n[1,])
## 
##  0  2  3  4  5  6 10 14 
##  2  6  5  2  4  1  4  1
barplot(table(nc$Best.n[1,]),
          xlab="Number of Clusters", ylab="Number of Criteria",
          main="Number of Clusters Chosen by 26 Criteria")

#We split up the data into 3 clusters above. The NbClust algorithm suggests 2 or 3 clusters as appropriate

cvhclust <- as.matrix(clusters)
cvpam <- as.matrix(pamcluster$clustering)
library(clValid)
hclust.cved <- clValid(cvhclust, 2:5, clMethods = c("hierarchical"), validation = "internal")
summary(hclust.cved)
## 
## Clustering Methods:
##  hierarchical 
## 
## Cluster sizes:
##  2 3 4 5 
## 
## Validation Measures:
##                                  2       3       4       5
##                                                           
## hierarchical Connectivity   4.2869  7.2159  9.2159 10.2159
##              Dunn           1.0000     Inf     NaN     NaN
##              Silhouette     0.9706  0.9804  0.9216  0.9216
## 
## Optimal Scores:
## 
##              Score  Method       Clusters
## Connectivity 4.2869 hierarchical 2       
## Dunn            Inf hierarchical 3       
## Silhouette   0.9804 hierarchical 3
pam.cved <- clValid(cvpam, 2:5, clMethods = c("pam"), validation = "internal")
summary(pam.cved)
## 
## Clustering Methods:
##  pam 
## 
## Cluster sizes:
##  2 3 4 5 
## 
## Validation Measures:
##                         2       3       4       5
##                                                  
## pam Connectivity   0.0000  4.3825  7.3115 10.2405
##     Dunn           1.0000     Inf     NaN     NaN
##     Silhouette     0.8919  1.0000  0.7451  0.7451
## 
## Optimal Scores:
## 
##              Score Method Clusters
## Connectivity   0   pam    2       
## Dunn         Inf   pam    3       
## Silhouette     1   pam    3
#Since PAM has a lower Connectivity and higher Silhouette, it seems like that is a better clustering fit to the data. Also, looking at the various clusters, it does seem like 3 clusters is the optimal number of clusters. 

RYAN ———————————————————————-

Creating a dataset with no clusters:

Create a bivariate distribution with function rnorm2d(). Below is a 3D representation of a bivariate distribution. Since the means are centered in the 2D space, there are no clusters to be found.

Bivariate_Plot

Bivariate_Plot

library(fMultivar)
## Loading required package: timeDate
## Loading required package: timeSeries
## Loading required package: fBasics
## 
## Rmetrics Package fBasics
## Analysing Markets and calculating Basic Statistics
## Copyright (C) 2005-2014 Rmetrics Association Zurich
## Educational Software for Financial Engineering and Computational Science
## Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY.
## https://www.rmetrics.org --- Mail to: info@rmetrics.org
## 
## Attaching package: 'fBasics'
## The following object is masked from 'package:flexclust':
## 
##     getModel
## The following object is masked from 'package:modeltools':
## 
##     getModel
## 
## Rmetrics Package fMultivar
## Analysing and Modeling Multivariate Financial Return Distributions
## Copyright (C) 2005-2014 Rmetrics Association Zurich
## Educational Software for Financial Engineering and Computational Science
## Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY.
## https://www.rmetrics.org --- Mail to: info@rmetrics.org
library(dplyr)
set.seed(1234)
df <- rnorm2d(1000, rho=.5) %>% as.data.frame()
plot(df, main="Bivariate Normal Distribution with rho=0.5")

Looking for clusters:

  • From the within-groups sum of squares plot, we see that the analysis suggests 2 or 3 clusters in the data.

  • Using NbClust() and specifying the clustering method as “kmeans”, the result suggests the data has 2 clusters in it.

wssplot <- function(data, nc=15, seed=1234){
               wss <- (nrow(data)-1)*sum(apply(data,2,var))
               for (i in 2:nc){
                    set.seed(seed)
                    wss[i] <- sum(kmeans(data, centers=i)$withinss)}
                plot(1:nc, wss, type="b", xlab="Number of Clusters",
                     ylab="Within groups sum of squares")}
wssplot(df)

library(NbClust)
#nc <- NbClust(df, min.nc=2, max.nc=15, method="kmeans")
#saveRDS(nc, file = "nc.rds")
nc <- readRDS(file = "Files/Ryan/nc.rds")

barplot(table(nc$Best.n[1,]),
        xlab="Number of Clusters", ylab="Number of Criteria",
        main="Number of Clusters Chosen by 26 Criteria")

Visualizing the suggested clusters:

  • Using pam(), part of the {cluster} package, we partition the data into the two clusters suggested by the previous analysis. The technique pam() is short for partitioning around medoids. The documentation says it is a “more robust version of K-means”.
library(ggplot2)
library(cluster)

fit <- pam(df, k=2) # fit the clusters
df$clustering <- fit$clustering %>% factor() # add the cluster tags to the main DF


ggplot(data=df, aes(x=V1, y=V2, color=clustering, shape=clustering)) +
  geom_point() +
  ggtitle("Clustering of Bivariate Normal Data")

How to remedy the problem

Kabakoff suggests the “Cubic Clustering Criteria” (CCC), part of the NbClust report, but first gives the disclaimer that “it isn’t foolproof”.

In a broad overview, CCC compares the R^2 you get with a given number of clusters with the R^2 you would get with a uniform distribution. So, the largest difference is what you are after.

  • With CCC, the best number of clusters will correspond to a local maxima of the CCC index.

  • If you see a continuously decreasing CCC, then it suggests there are no clusters present.

In summary, analysis of the results from NbClust looking at within sum of squares and NbClusts’ “26 criteria” suggested 2 clusters were present. The indicator that strongly suggested otherwise was the CCC.

In cluster analysis and with all modeling, it is important to not overfit your data and use your intuition and knowledge of the data to deduce what is and is not trend.

plot(nc$All.index[,4], type="o", ylab="CCC",
            xlab="Number of clusters", col="blue")